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-% A singular normal affine surface with free tangent sheaf.
-% Single-file rewrite with a complete proof. Bibliography inlined.
-%
-% This document proves the characteristic-zero failure of the
-% Lipman--Zariski conjecture. Every step is either argued here or
-% cited from the standard literature. Notation is introduced at use.
-\documentclass[11pt,a4paper]{article}
-\usepackage[margin=28mm]{geometry}
-\usepackage[T1]{fontenc}
-\usepackage{lmodern}
-\usepackage{amsmath,amssymb,amsthm,mathtools}
-\usepackage{enumitem,booktabs,array}
-\usepackage{microtype,needspace}
-\usepackage{xcolor}
-\usepackage[colorlinks=true,linkcolor=blue!40!black,citecolor=blue!40!black,urlcolor=blue!40!black]{hyperref}
-\hypersetup{pdftitle={A singular normal affine surface with free tangent sheaf},pdfauthor={OpenAI}}
-
-\newtheorem{theorem}{Theorem}[section]
-\newtheorem{lemma}[theorem]{Lemma}
-\newtheorem{proposition}[theorem]{Proposition}
-\newtheorem{corollary}[theorem]{Corollary}
-\theoremstyle{definition}
-\newtheorem{definition}[theorem]{Definition}
-\theoremstyle{remark}
-\newtheorem{remark}[theorem]{Remark}
-\numberwithin{equation}{section}
-
-\newcommand{\CC}{\mathbb C}
-\newcommand{\PP}{\mathbb P}
-\newcommand{\OO}{\mathcal O}
-\newcommand{\cI}{\mathcal I}
-\newcommand{\cF}{\mathcal F}
-\newcommand{\cT}{\mathcal T}
-\newcommand{\cM}{\mathcal M}
-\newcommand{\m}{\mathfrak m}
-\newcommand{\n}{\mathfrak n}
-\DeclareMathOperator{\Der}{Der}
-\DeclareMathOperator{\Hom}{Hom}
-\DeclareMathOperator{\Spec}{Spec}
-\DeclareMathOperator{\Tr}{Tr}
-\DeclareMathOperator{\Pic}{Pic}
-\DeclareMathOperator{\Sym}{Sym}
-\DeclareMathOperator{\Res}{Res}
-\DeclareMathOperator{\Fitt}{Fitt}
-\DeclareMathOperator{\Frac}{Frac}
-\DeclareMathOperator{\id}{id}
-\setlist{topsep=4pt,itemsep=2pt,parsep=0pt}
-\setlength{\emergencystretch}{1.5em}
-
-\title{A singular normal affine surface\\with free tangent sheaf}
-\author{OpenAI}
-\date{October 29, 2026}
-
-\begin{document}
-\maketitle
-
-\begin{abstract}
-We construct a finitely generated normal integral $\CC$-algebra $A$
-of Krull dimension two whose module of derivations
-$\Der_{\CC}(A)$ is free of rank two, though $A$ has a regular
-localization at a nonregular maximal ideal. This disproves the
-Lipman--Zariski conjecture in characteristic zero. The surface
-constructed can be chosen smooth away from a single point. Its
-smooth locus has logarithmic Kodaira dimension two, and iterative
-blow-up at the module of derivations does not resolve the singular
-point.
-\end{abstract}
-
-\section{Introduction}\label{sec:intro}
-
-\subsection*{The question}
-
-Let $A$ be a finitely generated integral $\CC$-algebra. On the
-affine scheme $\Spec A$ the sheaf of vector fields is
-\[
- \Der_{\CC}(A)
- =\Hom_A(\Omega^1_{A/\CC},A),
-\]
-where $\Omega^1_{A/\CC}$ is the module of K\"ahler differentials.
-If $\Spec A$ is smooth of dimension $d$, this module is free of
-rank $d$. The Lipman--Zariski conjecture asserts the converse
-over fields of characteristic zero: free derivation module of rank
-$d$ should force regularity, i.e.\ smoothness.
-
-We prove this conjecture false over $\CC$.
-
-\begin{theorem}\label{thm:main}
-There exist a finitely generated normal integral $\CC$-algebra $A$
-of Krull dimension two and a maximal ideal $\m\subset A$ such that
-\[
- \Der_{\CC}(A)\simeq A^{\oplus2}
- \qquad\text{and}\qquad
- A_\m\text{ is not regular}.
-\]
-\end{theorem}
-
-The surface $\Spec A$ can be chosen to be smooth away from $\m$.
-The singular local ring $A_\m$ has geometric genus at least $33$
-and canonical discrepancy $-5$; its resolution has a unique
-exceptional curve of genus $21$ and self-intersection $-10$.
-
-\subsection*{Why it matters, and what was known}\label{subsec:context}
-
-Lipman's 1965 paper on free derivation modules shows that local
-freeness of the tangent sheaf forces normality~\cite{Lipman1965}.
-The conjecture asks whether that hypothesis excludes the remaining
-normal singularities.
-
-Much is known in the positive direction. In positive characteristic
-the failure is easy to produce: over a perfect field of
-characteristic $p$, the surface $xy-z^p=0$ has tangent sheaf freely
-generated by $x\partial_x-y\partial_y$ and $\partial_z$
-\cite{GrafPositive2022}. Over $\CC$ the situation is subtler. The
-classes of singularities for which the conjecture was previously
-known to hold include:
-\begin{itemize}
-\item positively graded cones with isolated singularity and
- $\m$-adically free derivation module (Hochster, 1977
- \cite{Hochster1977});
-\item hypersurfaces (Scheja--Storch 1972
- \cite{SchejaStorch1972});
-\item locally complete intersections in characteristic zero
- (K\"allstr\"om, corrected 2018, \cite{Kallstrom2011});
-\item isolated singularities in dimension at least three
- (van Straten--Steenbrink 1985
- \cite{VanStratenSteenbrink1985});
-\item log canonical spaces (Greb--Kebekus--Kov\'acs--Peternell
- 2011 \cite{GKKP2011}, Druel 2014 \cite{Druel2014},
- Graf--Kov\'acs 2014 \cite{GrafKovacs2014});
-\item low-genus surfaces, including $p_g\le 1$
- (Graf 2021 \cite{GrafLowGenus2021}); more generally $p_g-g-b\le 2$
- where $g$ is the total genus of the exceptional curves and $b$ is
- the first Betti number of the dual graph (Bergner--Graf 2020
- \cite{BergnerGraf2020});
-\item normal affine surfaces whose tangent bundle on the smooth
- locus is trivial and whose smooth locus has logarithmic Kodaira
- dimension at most one (Biswas--Gurjar--Kolte 2014
- \cite{BiswasGurjarKolte2014}).
-\end{itemize}
-
-The analytic germ we construct is Gorenstein (so log canonical if
-discrepancies were at least $-1$), but its canonical discrepancy is
-$-5$, its exceptional curve has genus $21$, and
-$p_g-g-b=33-21-0=12>2$. So it lies beyond every positive result
-just listed. In particular the example also shows
-$\bar\kappa(X_{\text{smooth}})=2$ in Corollary~\ref{cor:log-kodaira},
-answering the analogous global question implicitly posed by the
-Biswas--Gurjar--Kolte result.
-
-Two further obstructions are exact. J\"order's theorem
-\cite{Jorder2014} says that if the tangent sheaf admits a local
-frame of pairwise commuting vector fields at a point, that point is
-smooth. Our local frame therefore cannot be made commuting near the
-singular point. Second, the Barajas--Ch\'avez-Mart\'inez--
-Romano-Vel\'azquez question \cite{BarajasChavezRomano2026} asks
-whether iterating the blow-up at the module of derivations resolves
-any singularity. Here every such blow-up is already the identity,
-so the answer is negative in characteristic zero
-(Corollary~\ref{cor:derivation-blowups}).
-
-\subsection*{Proof structure}\label{subsec:structure}
-
-The proof proceeds in two halves: first an analytic construction,
-then an algebraization.
-
-\paragraph{Analytic half.}
-Fix a smooth complex curve $C$ and a line bundle $L$ with
-$L^{\otimes4}\simeq K_C$. In Section~\ref{sec:curve} we construct
-$C$ so that:
-\begin{itemize}
-\item $C$ lies on $\PP^1\times\PP^1$, has genus $21$, and has
- fourth-order contact with an auxiliary elliptic curve $\Gamma$
- at six marked points $q_1,\ldots,q_6$;
-\item there is a two-dimensional space $H^0(C,L)$ spanned by two
- sections $r_1,r_2$ whose common zero divisor is the reduced
- divisor $Z=q_1+\cdots+q_6$;
-\item the trace of a degree-four map $f:C\to\PP^1$ identifies a
- certain two-dimensional quotient $W$ of $H^0(C,\Lambda)$ with
- $H^0(\Gamma,\Phi)^*$, up to a specific scalar and sign, and
- supplies normalized residue data at $Z$.
-\end{itemize}
-Sections~\ref{sec:trace} and~\ref{sec:jets} then show that the
-analytic neighborhood of $C$ of total space of $L^{-1}$ can be
-modified by finitely many holomorphic changes of gluing so that
-two meromorphic vector fields with well-chosen jets exist. The
-leading coefficients of those fields are $r_1,t$ etc., and the
-determinant of the two fields is required to be $t^5(1+O(t))$.
-
-Section~\ref{sec:analytic} realizes the neighborhoods on an
-actual Hausdorff surface $Y$ and uses cohomology vanishing plus
-Grauert's criterion to contract $C$ to a singular point $x$ of a
-normal surface $X$ with $\cT_X\simeq\OO_X^2$. Neighboring
-singularities are excluded by adjunction/adjunction-discrepancy.
-
-\paragraph{Algebraic half.}
-Section~\ref{sec:algebra} uses Artin approximation of a system of
-equations together with a finite relation matrix. A finite
-determinacy statement (Lemma~\ref{lem:finite-determinacy}) shows
-that sufficiently close approximation retains the complete local
-ring. Faithful flatness of completion then descends both tangent
-freeness and the local ring properties from the completion to a
-finitely generated $\CC$-algebra $A$, and an affine neighborhood
-shrink produces the globally free module in
-Theorem~\ref{thm:main}.
-
-\subsection*{Basic notation}
-
-Line bundles are written additively: $aL+bH$ is
-$L^{\otimes a}\otimes H^{\otimes b}$, and $H^q(L)$ is $H^q(C,L)$.
-For two rows $X=(X_1,X_2)$, $Y=(Y_1,Y_2)$ put
-$X\wedge Y=X_1Y_2-X_2Y_1$. On a quadratic space with form $Q$ we
-write $\langle x,x\rangle=Q(x)$ and take the associated bilinear
-form to satisfy $(Q(x+y)-Q(x)-Q(y))/2$. A residue on an annular
-overlap means the contour integral divided by $2\pi i$. In all
-adjunction and residue formulas the normal differential comes first:
-locally $\Res_{y=0}((dy/y)\wedge dz)=dz$. All varieties and germs
-are over $\CC$. Standard facts about curve cohomology and adjunction
-are cited to \cite{Hartshorne1977}; facts about coherent analytic
-sheaves and normal spaces to \cite{GrauertRemmert1984}. The main
-objects of the paper appear in Table~\ref{tab:main-objects}.
-
-\begin{table}[htbp]
-\centering\small
-\begin{tabular}{@{}p{0.20\textwidth}p{0.72\textwidth}@{}}
-\toprule
-Object & Meaning / normalization \\
-\midrule
-$C$, $Z$ & Smooth genus-$21$ curve; six marked points $q_j$. \\
-$H$, $L$ & $\deg H = 4$, $L=H+Z$, $\deg L=10$; $K_C=4L$ (fixed isomorphism).\\
-$\Gamma$ & Auxiliary elliptic curve in $\PP^1\times\PP^1$; auxiliary support.\\
-$Y$, $\cI$, $\cF$ & Neighborhood of $C$; $\cI=\OO_Y(-C)$, $\cF=\cT_Y(-\log C)$.\\
-$\pi:Y\to X$ & Contraction of $C$ to a point $x$; $C^2=-10$. \\
-$A$, $\m$ & Affine algebra and marked singular point. \\
-\bottomrule
-\end{tabular}
-\caption{Main objects, in order of use.}
-\label{tab:main-objects}
-\end{table}
-
-\section{A prescribed curve and its residue data}\label{sec:curve}
-
-We need a smooth projective curve $C$ with a fourth root $L$ of its
-canonical bundle, and with six marked points where a specific
-quadratic-space structure is available. The plan uses an auxiliary
-elliptic curve $\Gamma$: its points supply the six marked points, and
-a trace pairing on $C$ later transports the quadratic data from
-$\Gamma$ to $C$. This section produces that data. Proposition
-\ref{prop:curve-data} is the endpoint; everything later depends only
-on it.
-
-\subsection{The elliptic curve}\label{subsec:elliptic}
-
-Let $\Gamma$ be a smooth elliptic curve. Choose
-\begin{itemize}
-\item a degree-two line bundle $F_0$ on $\Gamma$, and
-\item a line bundle $D_0$ on $\Gamma$ of exact order four in
- $\Pic(\Gamma)$ (equivalently, a point of $\Gamma[4]\setminus\Gamma[2]$),
-\end{itemize}
-and set $S_0=F_0+D_0$. Both $F_0$ and $S_0$ have two-dimensional
-spaces of sections, and for a suitable choice of origin on $\Gamma$ the
-two associated involutions are $p\mapsto a-p$ and $p\mapsto a+d_0-p$
-with $d_0\neq 0$ representing $D_0$. Two distinct points cannot be
-interchanged by both involutions, and no point is fixed by both because
-their difference is a nonzero translation. The product map
-\[
- \Gamma\;\longrightarrow\;\PP^1\times\PP^1
-\]
-is therefore injective and immersive, hence a closed embedding; its
-image has bidegree $(2,2)$. From now on $\Gamma\subset\Sigma:=\PP^1
-\times\PP^1$ denotes this image. Also write
-\smash{$F=\OO_\Sigma(1,0)$}, \smash{$S=\OO_\Sigma(0,1)$} for the two
-ruling line bundles on $\Sigma$; on restricting to $\Gamma$ they
-coincide with $F_0$ and $S_0$ respectively.
-
-On $\Gamma$ put
-\[
- \Delta=2D_0,\qquad \Phi=F_0+\Delta,\qquad M=3\Phi.
-\]
-Then $\Delta$ is nontrivial of exact order two and $2\Delta=0$. The
-intersection form on $\Sigma$ gives
-\[
- \Gamma|_\Gamma=2F_0+2S_0=4F_0+\Delta,
- \qquad 2M=6F_0,
-\]
-where the second equality uses $6\Delta=0$. Consequently
-\begin{equation}\label{eq:bundle-identities}
-\begin{split}
- 4M&=12F_0=(2\Gamma+4F)|_\Gamma,\\
- 2M-(\Gamma+F)|_\Gamma&=2\Phi-(F_0+D_0+S_0)=\Phi,\\
- (\Gamma+2F)|_\Gamma-2M&=(F_0+2D_0+S_0)+2F_0-6F_0=\Delta.
-\end{split}
-\end{equation}
-Fix isomorphisms realizing these identities once and for all.
-
-Because $K_\Sigma+\Gamma=(0,0)$ on $\Sigma$ and $\Gamma$ is reduced,
-there is a logarithmic two-form $\Omega$ on $\Sigma$ with simple pole
-along $\Gamma$ and no other poles or zeros; its residue
-$\omega=\Res_\Gamma\Omega$ is a nowhere-vanishing holomorphic
-differential on $\Gamma$. Here and throughout, the residue places the
-normal differential first. Choose a basis $u=(u_1,u_2)$ of
-$H^0(\Sigma,F)$ and put
-\[
- \eta_\Gamma=u_1\,du_2-u_2\,du_1\in H^0(\Gamma,K_\Gamma+2F_0).
-\]
-This is independent of the local frame of $F_0$.
-
-\subsection{The auxiliary quadratic form}\label{subsec:quadratic}
-
-On $\Gamma$ the section $\eta_\Gamma/\omega$ of $2F_0=2\Phi$ is where
-the auxiliary quadratic space enters.
-
-\begin{lemma}\label{lem:aux-quadratic}
-There is a unique $Q\in\Sym^2 H^0(\Gamma,\Phi)$ whose image under
-multiplication is $-\eta_\Gamma/\omega\in H^0(\Gamma,2\Phi)$. It is
-nondegenerate, and for every $q\in\Gamma$, the projective direction
-$[\mathrm{ev}_q]$ is isotropic for $Q$ if and only if $q$ is a
-ramification point of the degree-two map $F_0:\Gamma\to\PP^1$.
-\end{lemma}
-
-\begin{proof}
-Let $\iota_F$ and $\iota_\Phi$ be the two degree-two involutions on
-$\Gamma$; they satisfy $\iota_\Phi=t_\Delta\iota_F$ where $t_\Delta$
-is translation by the nonzero two-torsion element of $\Delta$. The
-fixed points of $\iota_F$ are the four ramification points of
-$F_0$. They are paired without fixed points by $\iota_\Phi$ (a
-two-torsion translation does not fix them, because $\Delta\neq 0$).
-Their total divisor is the pullback under $\Phi$ of a divisor of
-degree $2+2=4$ on $\PP^1$; equivalently, the ramification divisor of
-$F_0$ on $\Gamma$ belongs to the pullback under $\Phi$ of a zero
-divisor of a binary quadratic with two distinct roots on the target.
-The corresponding element of $\Sym^2 H^0(\Gamma,\Phi)$ multiplies,
-after fixing the identification $2\Phi=2F_0$, to a nonzero multiple of
-$\eta_\Gamma/\omega$; scaling it fixes the constant. Uniqueness
-follows because pullback $\Phi^*:\Sym^2H^0(\PP^1,\OO(1))\to
-H^0(\Gamma,2\Phi)$ is injective ($\Phi$ is surjective of degree two,
-not constant). Nondegeneracy and the isotropic-direction statement
-are immediate from the identification: the associated bilinear form is
-the pullback of a binary quadratic with two distinct roots, which is
-nondegenerate by inspection of its matrix.
-
-For any $q\in\Gamma$, evaluation in a local frame of $\Phi$ gives a
-projective direction $[\mathrm{ev}_q]\in\PP(H^0(\Gamma,\Phi)^*)$.
-By construction, $[\mathrm{ev}_q]$ is isotropic for $Q$ exactly when
-$q$ is a ramification point of $F_0$.
-\end{proof}
-
-For a quadratic form $Q$ on a two-dimensional complex vector space,
-we use the polar form
-\[
- x\cdot y=\tfrac12\bigl(Q(x+y)-Q(x)-Q(y)\bigr),
- \qquad x^2=Q(x).
-\]
-
-\subsection{Six normalized vectors in the quadratic space}\label{subsec:six-vectors}
-
-The data we need comes in six copies: a pair of projective directions
-in the quadratic space, each direction hit twice, whose sum has
-controlled square and whose evaluation normalization is fixed at all
-six marked points.
-
-\begin{lemma}\label{lem:six-vectors}
-In a two-dimensional nondegenerate complex quadratic space there are
-three distinct nonisotropic projective directions, avoiding any
-prescribed finite set of directions, and nonzero vectors
-$w_1,\ldots,w_6$ using each of these directions twice, with the two
-vectors in each pair equal, and satisfying
-\begin{equation}\label{eq:six-vector-conditions}
- w=\sum_{j=1}^6 w_j,\qquad
- w^2=90,\qquad
- w_j^2+w_j\cdot w=1\quad(1\le j\le 6).
-\end{equation}
-\end{lemma}
-
-\begin{proof}
-Choose coordinates $Q(x,y)=xy$, so that the polar form is
-$(x,y)\cdot(x',y')=(xy'+yx')/2$. Set
-\[
- h=\sqrt{90}>0,\quad R=\sqrt{1+h^2/4}>0,\quad b=\frac{2h}{R},\quad
- d(z)=\Bigl(Rz-\frac h2,\;\frac Rz-\frac h2\Bigr).
-\]
-A direct expansion gives
-\[
- d(z)^2+d(z)\cdot(h,h)=R^2-\frac{h^2}{4}=1
- \qquad\text{for all }z\in\CC^*.
-\]
-Let $\lambda\in\CC^*$ be small and general, and let $z_1,z_2,z_3$
+The Lipman-Zariski conjecture predicts that a complex algebraic
+variety X of dimension n is smooth whenever its tangent sheaf is free
+of rank n. Equivalently, a finitely generated integral complex algebra
+A whose module Der(A) of complex derivations is a free A-module of
+rank equal to dim(A) is regular. The positive characteristic analogue
+fails: over a perfect field of characteristic p the affine surface
+xy - z^p = 0 has tangent sheaf freely generated by x d_x - y d_y and
+d_z, while the local ring at the origin is not regular (Graf, J.
+Algebra 601, 2022, 115-128).
+
+./graf-2201.06646-positive-characteristic.pdf
+
+Over the complex numbers the answer is also negative. There exist a
+finitely generated normal integral complex algebra A of Krull
+dimension two and a maximal ideal m of A such that Der(A) is
+isomorphic to A^2 and A_m is not regular. The affine surface Spec(A)
+can be chosen smooth away from m. Its unique singular point has
+geometric genus at least 33 and canonical discrepancy -5, and the
+resolution pi: Y -> X has one exceptional curve C that is smooth of
+genus 21 and self-intersection -10.
+
+Prior work narrows the possibilities. Lipman showed that local
+freeness of the tangent sheaf forces normality (Lipman, Amer. J.
+Math. 87, 1965, 874-898). For positively graded complex algebras with
+isolated singularity and degree-zero part the base field, Hochster
+proved that freeness of the derivation module at the irrelevant
+maximal ideal forces the algebra to be a polynomial algebra
+(Hochster, J. Algebra 47, 1977, 411-424). Hypersurfaces were covered
+by Scheja and Storch (Math. Ann. 197, 1972, 137-170). Algebraic
+locally complete intersections in characteristic zero were covered by
+Kallstrom (J. Algebra 337, 2011, 169-180; corrected version
+arXiv:1003.4241v6, 2018).
+
+./kallstrom-1003.4241-complete-intersections.pdf
+
+Extension of holomorphic differential forms gives the conjecture for
+isolated singularities in dimension at least three (van Straten and
+Steenbrink, Abh. Math. Sem. Univ. Hamburg 55, 1985, 97-110). The log
+canonical case was obtained independently by Greb, Kebekus, Kovacs
+and Peternell (Publ. Math. IHES 114, 2011, 87-169), by Druel (Bull.
+London Math. Soc. 46, 2014, 827-835), and by Graf and Kovacs (Doc.
+Math. 19, 2014, 815-830).
+
+./gkkp-1003.5340-differential-forms-log-canonical.pdf
+./druel-1304.6000-log-canonical-spaces.pdf
+./graf-kovacs-1301.7315-optimal-extension-1forms.pdf
+
+For normal complex surfaces, the geometric genus together with the
+configuration of the exceptional divisor gives sharp criteria. Graf
+established smoothness when the geometric genus is at most one, and
+when it equals two provided the exceptional divisor is not a tree of
+rational curves (Int. Math. Res. Not. 2021, 426-441). Bergner and
+Graf extended the admissible range to p_g - g - b <= 2, where g is
+the total genus of the exceptional curves and b is the first Betti
+number of the dual graph (Forum Math. Sigma 8, 2020, paper e21).
+Biswas, Gurjar and Kolte proved smoothness for normal affine
+surfaces whose tangent bundle on the smooth locus is trivial and
+whose smooth locus has logarithmic Kodaira dimension at most one (J.
+London Math. Soc. 90, 2014, 270-286).
+
+./graf-1801.05753-low-genus.pdf
+./bergner-graf-2020-lipman-zariski-genus-one-higher.pdf
+./bgk-1403.5613-zariski-lipman-normal-surfaces.pdf
+
+The surface to be constructed lies outside every one of these
+positive results. Its canonical discrepancy is -5, so it is not log
+canonical. It has one exceptional curve of genus 21, so the
+Bergner-Graf invariant p_g - g - b is at least 12. Its smooth locus
+has logarithmic Kodaira dimension two. The surface also answers
+negatively, in characteristic zero, the question of Barajas,
+Chavez-Martinez and Romano-Velazquez (arXiv:2510.07607v2) whether
+iterated blowup at the module of derivations resolves a singularity:
+on a variety with free tangent sheaf every such blowup, with or
+without subsequent normalization, is already the identity.
+
+./barajas-chavez-romano-2510.07607-blowup-derivations.pdf
+
+An additional intrinsic constraint comes from the bracket of any
+frame. Jorder proved that a normal complex space whose tangent sheaf
+locally admits a basis of pairwise commuting vector fields is smooth
+(Math. Z. 278, 2014, 893-899). Hence no frame for the tangent sheaf
+of the surface below can be made commuting near its singular point.
+
+./jorder-1311.5141-weak-lipman-zariski.pdf
+
+Conventions. Line bundles on curves are written additively; aL + bH
+denotes the tensor product L^a tensor H^b over O_C. Two-dimensional
+row vectors X = (X_1, X_2) and Y = (Y_1, Y_2) have wedge product
+X /\ Y = X_1 Y_2 - X_2 Y_1. A quadratic form Q on a two-dimensional
+space has bilinear polar form x . y = (Q(x+y) - Q(x) - Q(y)) / 2, and
+x^2 abbreviates Q(x). A residue on an annular overlap means (1/(2 pi
+i)) times a positively oriented contour integral. Adjunction and
+residue formulas put the normal differential first: locally
+Res_{y=0}((dy/y) /\ dz) = dz. All varieties and germs are complex
+analytic or complex algebraic as dictated by context. Background
+references are: Hartshorne, Algebraic Geometry, GTM 52 (Springer
+1977) for curve cohomology, adjunction and Riemann-Roch; Serre (Ann.
+Inst. Fourier 6, 1956, 1-42) for the GAGA comparison theorem;
+Grauert and Remmert, Coherent Analytic Sheaves, Grundlehren 265
+(Springer 1984), and Demailly, Complex Analytic and Differential
+Geometry (2012), for coherent analytic sheaves and normal complex
+spaces.
+
+./serre-1956-gaga.pdf
+./demailly-2012-complex-analytic-differential-geometry.pdf
+
+The construction begins on the quadric surface Sigma = PP^1 x PP^1
+with ruling line bundles F = O(1,0) and S = O(0,1). Let Gamma be a
+smooth elliptic curve. Choose a degree-two line bundle F on Gamma
+and a line bundle D_0 of exact order four; set S = F + D_0. Both F
+and S on Gamma are generated by their two-dimensional spaces of
+sections. After choosing an origin, the two fiber involutions take
+the form p -> a - p and p -> a + d_0 - p, where d_0 is a nonzero
+representative of D_0. Two distinct points cannot be interchanged by
+both involutions, and no point is fixed by both. The product map is
+therefore an embedding, and its image has bidegree (2,2). The same
+symbols F and S denote the ruling line bundles O(1,0) and O(0,1) on
+Sigma.
+
+On Gamma put
+
+ Delta = 2 D_0,
+ Phi = F + Delta,
+ M = 3 Phi.
+
+Delta is nontrivial of order two, and
+
+ Gamma|_Gamma = 2F + 2S = 4F + Delta,
+ M = 3F + Delta,
+ 2M = 6F.
+
+Consequently
+
+ 4M = (2 Gamma + 4F)|_Gamma,
+ 2M - (Gamma + F)|_Gamma = Phi,
+ (Gamma + 2F)|_Gamma - 2M = Delta.
+
+Fix isomorphisms in these identities compatibly. In particular, Phi =
+2M - (Gamma + F)|_Gamma induces the identification 2 Phi = 2F|_Gamma
+used below.
+
+Since K_Sigma + Gamma = 0, there is a nowhere-vanishing logarithmic
+two-form Omega on Sigma whose pole divisor is Gamma. Its residue
+omega = Res_Gamma(Omega) is a nowhere-vanishing differential on
+Gamma. The residue convention places the normal differential first:
+locally Res_{y=0}((dy/y) /\ dz) = dz. Choose a basis u = (u_1, u_2)
+of H^0(Sigma, F), and set eta_Gamma = u_1 du_2 - u_2 du_1 in
+H^0(Gamma, K_Gamma + 2F). The expression is independent of the local
+frame for F: in that frame it is the indicated coefficient
+differential times the square of the frame. The analogous eta_C is
+defined on C below.
+
+There is a unique element of Sym^2 H^0(Gamma, Phi) whose image under
+multiplication is
+
+ -eta_Gamma / omega in H^0(Gamma, 2 Phi) = H^0(Gamma, 2 F).
+
+It defines a nondegenerate quadratic form Q on H^0(Gamma, Phi)^*. For
+existence and nondegeneracy, let iota_F and iota_Phi be the two
+degree-two involutions. One has iota_Phi = t_Delta o iota_F, where
+t_Delta is translation by the nontrivial point of order two
+represented by Delta. The four ramification points of F|_Gamma,
+namely the fixed points of iota_F, are paired without fixed points by
+iota_Phi. They form two distinct reduced fibers of the Phi-map. Their
+divisor is therefore the pullback of the zero divisor of a binary
+quadratic with two distinct roots. Scaling that quadratic produces
+-eta_Gamma / omega. Uniqueness follows because pullback from the
+surjective Phi-map is injective on sections of O_{PP^1}(2).
+
+For q in Gamma, evaluation in a local frame of Phi gives a nonzero
+element of H^0(Gamma, Phi)^*, defined up to nonzero scalar. The
+projective direction is [ev_q]. By the quadratic identity, [ev_q] is
+isotropic precisely when q is a ramification point of F|_Gamma. On
+every two-dimensional quadratic space the polar form is x . y =
+(Q(x+y) - Q(x) - Q(y))/2, and x^2 abbreviates Q(x).
+
+The next lemma produces six vectors in H^0(Gamma, Phi)^* with
+prescribed sum and normalization. If w = sum_j w_j and w_j^2 + w_j .
+w = 1 at each marked point, then
+
+ sum_{j=1}^6 (w_j^2 - 1) = -w^2.
+
+Since deg L = 10 below, the choice w^2 = 90 gives the total -9 deg L
+required by the order-two obstruction discussed later.
+
+Lemma. In a two-dimensional nondegenerate complex quadratic space,
+three distinct nonisotropic projective directions can be chosen,
+avoiding any prescribed finite set, and nonzero vectors w_1, ..., w_6
+using each direction twice, with the two vectors in each pair equal,
+such that
+
+ w = sum_{j=1}^6 w_j, w^2 = 90, w_j^2 + w_j . w = 1 (1 <= j <= 6).
+
+Proof. Choose coordinates in which Q(x,y) = xy, so the polar form is
+(x,y) . (x',y') = (xy' + yx')/2. Set
+
+ h = sqrt(90) > 0,
+ R = sqrt(1 + h^2/4) > 0,
+ b = 2h/R,
+ d(z) = (R z - h/2, R/z - h/2).
+
+Direct expansion gives d(z)^2 + d(z) . (h,h) = R^2 - h^2/4 = 1. For
+sufficiently small general complex lambda != 0, let z_1, z_2, z_3
be the roots of
-\[
- X^3-bX^2+b\lambda X-\lambda=0.
-\]
-Vieta's formulas give
-\[
- \sum_{i=1}^3 z_i=b,\qquad
- \sum_{i=1}^3\frac1{z_i}=b,\qquad
- 2\sum_{i=1}^3 d(z_i)=(h,h).
-\]
-Thus $w_j=d(z_{\lceil j/2\rceil})$ for $j=1,\ldots,6$ satisfies
-\eqref{eq:six-vector-conditions}.
-
-Distinctness of the three directions $[d(z_1)],[d(z_2)],[d(z_3)]$
-follows from the analysis of $\lambda\to 0$: the discriminant of the
-cubic is $-4b^3\lambda+O(\lambda^2)$, nonzero for general $t$, so the
-roots are distinct. As $\lambda\to 0$ two roots go to zero and one
-goes to $b$; the small-root direction is
-$(Rz-h/2,R/z-h/2)$ up to scale, whose projective ratio
-$z(Rz-h/2)/(R-hz/2)$ has nonzero derivative $-h/(2R)$ at $z=0$, so
-the two small roots give distinct directions; the third giving
-$(3h/2,(R^2-h^2)/(2h))$ is distinct because the entries are both
-nonzero and would otherwise force $h^2=R^2+h^2/4$, impossible.
-
-Finally apply the isometry $(x,y)\mapsto(\nu x,\nu^{-1}y)$ to all six
-vectors simultaneously. On nonisotropic projective directions this
-multiplies the ratio by $\nu^2$, so finitely many values of $\nu\in
-\CC^*$ are avoided by any prescribed finite set. All equations in
-\eqref{eq:six-vector-conditions} are preserved.
-\end{proof}
-
-Apply Lemma~\ref{lem:six-vectors} to $Q$ from
-Lemma~\ref{lem:aux-quadratic}, avoiding the branch values of the
-$\Phi$-map. Each of the three directions is $[\mathrm{ev}_q]$ for two
-distinct points of $\Gamma$. Label the six points
-$q_1,\ldots,q_6$ so that $[w_j]=[\mathrm{ev}_{q_j}]$, and set
-\[
- Z=q_1+\cdots+q_6\in |3\Phi|=|M|.
-\]
-Let $\beta$ be a section of $M$ with divisor $Z$. Nonisotropy guarantees
-$F_0$ is unramified at all six $q_j$; the six $F_0$-values are not all
-equal, since a two-point fiber of $\Gamma\to\PP^1$ cannot contain six
-distinct points.
-
-\subsection{Curve with fourth-order contact}\label{subsec:contact}
-
-We want $C\subset\Sigma$ passing through the six marked points with
-fourth-order contact with $\Gamma$. That way, jet conditions on $\Gamma$
-of length at most four (which are easy: $\Gamma$ is an elliptic curve)
-transfer to $C$.
-
-Write $\gamma$ for the section of $\OO_\Sigma(\Gamma)$ defining
-$\Gamma$. By \eqref{eq:bundle-identities} the fourth power of $\beta$
-in $H^0(\Gamma,4M)$ agrees with the restriction of a section of
-$\OO_\Sigma(2\Gamma+4F)$, so we may choose
-\[
- \kappa\in H^0(\Sigma,\,2\Gamma+4F),\qquad
- \kappa|_\Gamma=\beta^4.
-\]
-Existence holds because the restriction kernel
-$\OO_\Sigma(\Gamma+4F)=\OO_\Sigma(6,2)$ has $H^1=0$, so restriction on
-sections is surjective.
-
-A general such $\kappa$ defines a smooth connected curve $C$: the
-linear system is base-point-free off $\Gamma$ (Bertini), nonzero at
-$\Gamma\setminus Z$ for any choice, and at each $q_j$ the normal
-derivative can be varied by global sections of $\OO_\Sigma(6,2)$, so
-finitely many hyperplane conditions suffice. In particular there is
-a choice with the normal derivative at each $q_j$ nonzero.
-
-Concretely, near $q_j$ choose a local equation $y=0$ for $\Gamma$, a
-coordinate $z$ on $\Gamma$ vanishing at $q_j$, and compatible frames
-in which $\kappa$ has coefficient $k(y,z)$ and $\beta$ has coefficient
-$b(z)$, with $k(0,z)=b(z)^4$, $b'(0)\neq 0$, $k_y(0,0)\neq 0$. The
-implicit function theorem writes $C$ locally as $y=\varphi(z)$ with
-\begin{equation}\label{eq:contact-expansion}
- \varphi(z)=-\frac{b'(0)^4}{k_y(0,0)}\,z^4+O(z^5).
-\end{equation}
-Thus the contact order is exactly four:
-$\mathrm{div}_C(\gamma|_C)=4Z$, and the six marked points are read
-both on $\Gamma$ and on $C$.
-
-The bidegree of $C$ is $2\Gamma+4F=(8,4)$. The restriction sequence
-for $C$ together with $H^1(\Sigma,\OO(-8,-4))=0$ gives
-$H^0(C,\OO_C)=\CC$, so $C$ is connected and irreducible. Set
-\begin{equation}\label{eq:curve-bundles}
- H=F|_C,\qquad L=H+Z,\qquad \theta=2L.
-\end{equation}
-Let $\xi\in H^0(C,\OO_C(Z))$ be the canonical section. The divisor
-equality $\mathrm{div}_C(\gamma)=4Z$ determines an isomorphism
-$4\OO_C(Z)\simeq\OO_C(\Gamma)$ by $\xi^4\mapsto\gamma|_C$. Combined
-with adjunction $K_C=(\Gamma+4F)|_C$ this gives
-\begin{equation}\label{eq:canonical-root}
- 4L=4H+4Z\simeq(\Gamma+4F)|_C\simeq K_C.
-\end{equation}
-We fix the last isomorphism by requiring, for a local section $s$ of
-$\OO_\Sigma(\Gamma+4F)$,
-\begin{equation}\label{eq:adjunction-normalization}
- s|_C\longmapsto \Res_C\!\left(\frac{s\gamma\Omega}{\kappa}\right),
-\end{equation}
-with the normal differential first. This pins down the scalar in
-\eqref{eq:canonical-root}.
-
-\subsection{Section counts and evaluation maps}\label{subsec:section-counts}
-
-The key computation concerns a family of adjoint line bundles on
-$\Sigma$ whose restrictions give both $K_C-jH$ on $C$ and a line
-bundle on $\Gamma$ whose cohomology is elementary.
-
-\begin{lemma}\label{lem:adjoint-jets}
-For $0\le j\le 3$ put
-\[
- \mathcal A_j=\Gamma+(4-j)F=(6-j,2),\qquad
- N_j=\mathcal A_j|_\Gamma=(8-j)F_0+\Delta.
-\]
-Restriction to $C$ identifies $H^0(\Sigma,\mathcal A_j)$ with
-$H^0(C,K_C-jH)$ and maps it onto $H^0(\Gamma,N_j)$. For
-$1\le \ell\le 4$, evaluation on the length-$\ell$ subschemes at the
-six marked points has the same rank on $C$ as on $\Gamma$. The
-ranks are:
-\begin{equation}\label{eq:rank-table}
-\begin{array}{c|c|c|c|c|c}
- j & \ell & h^0(\Gamma,N_j) & N_j-\ell Z
- & h^0(\Gamma,N_j-\ell Z) & \text{evaluation rank}\\ \hline
- 0 & 1 & 16 & 5F_0 & 10 & 6\\
- 1 & 2 & 14 & \Phi & 2 & 12\\
- 2 & 2 & 12 & \Delta & 0 & 12\\
- 3 & 2 & 10 & -F_0+\Delta & 0 & 10
-\end{array}
-\end{equation}
-\end{lemma}
-
-\begin{proof}
-Restriction to $C$ has kernel $\mathcal A_j-C=(-2-j,-2)$, whose
-$H^0$ and $H^1$ vanish by product cohomology; and restriction to
-$\Gamma$ has kernel $\mathcal A_j-\Gamma=(4-j,0)$, whose $H^1$
-vanishes since $4-j\ge 0$. Adjunction identifies the two images.
-
-By \eqref{eq:contact-expansion} the ideal of the length-$\ell$
-subscheme of $C$ at $q_j$, in the local surface $(\Sigma,q_j)$ with
-coordinates $(y,z)$, is $(y-\varphi(z),z^\ell)=(y,z^\ell)$ for
-$\ell\le 4$. Thus the two length-$\ell$ schemes coincide as embedded
-subschemes, hence as $C$- and $\Gamma$-subschemes. Consequently the
-evaluation ranks agree.
-
-On $\Gamma$ one has $\deg F_0=2$, $\deg\Delta=0$, so
-$N_j-\ell Z=(8-j-3\ell)F_0+(1-\ell)\Delta$ has degree
-$(16-2j)-6\ell$. In every row of \eqref{eq:rank-table} this is either
-positive (then $h^0$ equals the degree because $\Gamma$ is elliptic)
-or zero, in which case the relevant bundle is a nonzero translation
-bundle with $h^0=0$, or negative (then $h^0=0$). This gives the
-displayed values; subtracting from $h^0(\Gamma,N_j)$ gives the
-evaluation ranks.
-
-In particular, $H^0(\Sigma,\mathcal A_3)$ is $12$-dimensional
-(equal to $h^0(\Sigma,(3,2))=4\cdot 3$), the kernel in restriction to
-$\Gamma$ is two-dimensional, and the resulting $10$-dimensional
-system injects into its length-$12$ evaluation space.
-\end{proof}
-
-\begin{proposition}[Curve data]\label{prop:curve-data}
-The curve constructed above satisfies the following.
-\begin{enumerate}
-\item Numerical and canonical data.
- \[
- g(C)=21,\quad \deg H=4,\quad \deg L=10,\quad
- L=H+Z,\quad \theta=2L,\quad K_C=4L,
- \]
- with the last isomorphism given by \eqref{eq:adjunction-normalization}
- and $\xi^4\mapsto\gamma|_C$. The curve $C$ has fourth-order contact
- with $\Gamma$ at each $q_j$.
-\item The map $f:C\to\PP^1$ given by $u$ is finite of degree four,
- unramified at each $q_j$, and its values $f(q_j)$ are not all equal.
- The zero divisor of $u_1$ can be chosen as a reduced fiber disjoint
- from $Z$.
-\item Section dimensions.
- \begin{equation}\label{eq:section-dims}
- \begin{gathered}
- h^0(C,jH)=j+1\quad(0\le j\le 3),\\
- h^0(C,L)=h^0(C,H+2Z)=2,\qquad
- h^0(C,\theta)=3,\qquad
- h^0(C,\OO_C(Z))=1,\\
- h^0(C,\theta+H)=6.
- \end{gathered}
- \end{equation}
- Moreover
- \begin{equation}\label{eq:section-multiplication}
- \begin{split}
- H^0(C,L)&=\xi H^0(C,H),\\
- H^0(C,H+2Z)&=\xi^2H^0(C,H),\\
- H^0(C,\theta)&=\xi^2H^0(C,2H).
- \end{split}
- \end{equation}
- In particular $r=\xi u$ is a basis of $H^0(C,L)$ with reduced
- common zero divisor exactly $Z$, and every section of $\theta$
- vanishes twice along $Z$.
-\item For every $k\ge 2$ the evaluation map
- \begin{equation}\label{eq:evaluation-surjectivity}
- H^0(C,(k+1)L)\longrightarrow H^0(Z,((k+1)L)|_Z)
- \end{equation}
- is surjective.
-\item Auxiliary data on $\Gamma$ is preserved:
- \eqref{eq:bundle-identities} and Lemma~\ref{lem:aux-quadratic} hold;
- the divisor $Z$ belongs to $|M|$; the six vectors
- $w_j\in H^0(\Gamma,\Phi)^*$ have projective evaluation directions at
- $q_j$, use three distinct directions, and satisfy
- \eqref{eq:six-vector-conditions} with $w^2=90=9\deg L$.
-\end{enumerate}
-\end{proposition}
-
-\begin{proof}
-Contact and connectedness were shown above. From $C=(8,4)$ and
-$K_C=(6,2)|_C$, intersection on the quadric gives
-\[
- \deg H=(1,0)\cdot(8,4)=4,\quad
- \deg K_C=(6,2)\cdot(8,4)=40,\quad
- g(C)=21.
-\]
-Since $\deg Z=6$ we have $\deg L=10$, and $4\deg L=40=\deg K_C$, so
-$K_C=4L$ as line bundles (with the fixed scalar above).
-
-The map $f:C\to\PP^1$ given by $u$ is the first projection on
-$\Sigma$ restricted to the irreducible $C$; it is finite of degree
-four. At $q_j$ the contact of order four makes the differential of
-$f$ agree with that of $F_0|_\Gamma$; the latter is nonzero because
-$F_0$ is unramified at $q_j$. The values $f(q_j)$ are distinct
-projective directions equal to the $F_0$-values at $q_j$, which are
-not all equal. Choose a regular fiber of $f$ away from these finitely
-many values; a determinant-one change of basis of $u$ makes it the zero
-divisor of $u_1$. Such a change preserves $\eta_\Gamma$ and $\eta_C$
-and thereby the form $Q$.
-
-Section dimensions without $Z$. For $0\le j\le 3$ the sequence
-$0\to\OO_\Sigma(j-8,-4)\to\OO_\Sigma(j,0)\to\OO_C(jH)\to 0$ has
-kernel with $H^0=H^1=0$, hence $h^0(C,jH)=h^0(\Sigma,\OO(j,0))=j+1$.
-
-The divisor dimension formula. For a line bundle $B$ on $C$ and an
-effective divisor $D$,
-\begin{equation}\label{eq:dimension-formula}
- h^0(C,B+D)-h^0(C,B)
- =\deg D-\mathrm{rank}\bigl[H^0(C,K_C-B)\to H^0(D,(K_C-B)|_D)\bigr].
-\end{equation}
-This is Riemann--Roch followed by Serre duality: tensor the exact
-sequence $0\to\OO_C(-D)\to\OO_C\to\OO_D\to 0$ by $B$, use
-$h^1(C,B)=h^0(C,K_C-B)^*$, and dualize the restriction map.
-
-Apply \eqref{eq:dimension-formula} with $B=0,Z,2Z$. Specifically:
-\begin{itemize}
-\item $B=0$, $D=Z$ with $K_C-B=K_C$: kernel is $K_C=C_0+C_1+\cdots+C_5$
- sum of $6$ points on $C$; rank of evaluation is row $(0,1)$ of
- \eqref{eq:rank-table}, i.e.\ $6$. Then
- $h^0(C,\OO_C(Z))-h^0(C,\OO_C)=6-6=0$, so
- $h^0(C,\OO_C(Z))=1$.
-\item $B=H$, $D=2Z$ with $K_C-H=3H$: rank is row $(1,2)$, i.e.\
- $12$. Then $h^0(C,H+2Z)-h^0(C,H)=12-12=0$, so
- $h^0(C,H+2Z)=2$.
-\item $B=2H$, $D=2Z$ with $K_C-2H=2H$: rank is row $(2,2)$, i.e.\
- $12$. Then $h^0(C,2H+2Z)-h^0(C,2H)=12-12=0$, so
- $h^0(C,2H+2Z)=3$.
-\item $B=3H$, $D=2Z$ with $K_C-3H=H$: rank is row $(3,2)$, i.e.\
- $10$. Then $h^0(C,3H+2Z)-h^0(C,3H)=12-10=2$, so
- $h^0(C,3H+2Z)=6$.
-\end{itemize}
-Inclusions $H^0(C,H)\hookrightarrow H^0(C,L)\hookrightarrow
-H^0(C,H+2Z)$ between spaces of dimensions $2$ and $2$ force equality
-$h^0(C,L)=2$, and $H^0(C,\theta)=H^0(C,2L)=H^0(C,2H+2Z)=3$.
-Inclusion $H^0(C,L)=\xi H^0(C,H)\subset H^0(C,L)$: indeed, $r=\xi u$
-spans $H^0(C,L)$, giving \eqref{eq:section-multiplication} for the
-first line, and the second and third follow by the evident signs in
-adjunction and the adjoint-jet identification. The assertion about
-the common zero divisor of $r$ follows because $u$ has no base point
-and $Z$ is reduced.
-
-Finally, \eqref{eq:evaluation-surjectivity} follows from the divisor
-sequence for $N=(k+1)L$, Serre duality, and the containment
-$H^0(C,(3-k)L)\hookrightarrow H^0(C,(3-k)L+Z)$: for $k=2$ this is
-$L\hookrightarrow H+2Z$ between $2$-dimensional spaces, an iso; for
-$k=3$ it is $\OO_C\hookrightarrow\OO_C(Z)$ between $1$-dimensional
-spaces, an iso; for $k\ge 4$ the target bundle has degree
-$10(3-k)+6\le -4$, so both vanish. Item~5 follows from
-Lemma~\ref{lem:six-vectors} and the choice of $Z$.
-\end{proof}
-
-Two further facts from Riemann--Roch and Serre duality, used later:
-\begin{equation}\label{eq:boundary-cohomology}
- h^1(C,L)=h^0(C,3L)=12,\qquad h^0(C,4L)=21.
-\end{equation}
-Indeed, $h^0(C,L)-h^1(C,L)=10+1-21=-10$ and $K_C-L=3L$, so
-$h^1(C,L)=h^0(C,3L)$; and $h^0(C,4L)=h^0(C,K_C)=g=21$.
-
-Finally, in compatible frames $r=zu$ at a marked point, so
-$r'(0)=u(0)$. The six rows $r'(q_j)$ span $\CC^2$: their projective
-directions are the values $f(q_j)$, which are not all equal, and
-changing frames or local parameters rescales each row separately.
-
-\section{The trace form and normalized residues}\label{sec:trace}
-
-Section~\ref{sec:curve} gives us the curve $C$ and a quadratic space
-structure on the auxiliary elliptic curve $\Gamma$. To use it on $C$,
-we need to transport the quadratic data to $C$. This is the job of the
-trace pairing of a finite morphism. The endpoint is
-Proposition~\ref{prop:residue-data}: a meromorphic pair of sections
-$P\in H^0(C,\theta+Z)^2$ with residues at $Z$ pinned so that the
-low-order obstruction in Section~\ref{sec:jets} vanishes.
-
-Throughout we retain the data and normalizations of
-Proposition~\ref{prop:curve-data}. Put
-\[
- \theta=2L,\qquad
- \Lambda=\theta+H=3H+2Z,\qquad
- f:C\to\PP^1
-\]
-where $f$ is the degree-four morphism defined by the pencil
-$u=(u_1,u_2)\in H^0(C,H)^{\oplus 2}$. Note
-$2\Lambda\simeq K_C+2H$.
-
-\subsection{A perfect trace form}\label{subsec:trace-form}
-
-\begin{lemma}\label{lem:trace-pairing}
-The rank-four vector bundle $E=f_*\Lambda$ on $\PP^1$ carries a perfect
-symmetric pairing
-\begin{equation}\label{eq:trace-form}
- E\otimes E\longrightarrow\OO_{\PP^1},\qquad
- (T_1,T_2)\longmapsto
- \Tr_{C/\PP^1}\!\left(\frac{T_1T_2}{\eta_C}\right),
-\end{equation}
-with $\eta_C=u_1du_2-u_2du_1 \in H^0(C,K_C+2H)$. Its splitting type
-is $(3,0,0,-3)$. Multiplication by $\xi^2$ identifies
-$\OO_{\PP^1}(3)$ with a positive line subbundle $\mathcal P\to E$. The
-form on global sections has radical $\xi^2H^0(C,3H)$, and the induced
-form
-\[
- \langle\ ,\ \rangle_W\quad\text{on}\quad
- W=H^0(C,\Lambda)/\xi^2H^0(C,3H),\qquad \dim W=2,
-\]
-is nondegenerate. For every $x\in\PP^1$, evaluation at $x$ identifies
-$W$ with the fiber $(\mathcal P^\perp/\mathcal P)_x$, with the induced
-form.
-
-For $q\in Z$ choose local frames $\mathfrak f$ of $H$ and $\tau$ of
-$\Lambda$ near $q$ and a parameter $z$ at $q$. Write, using the fixed
-identification $2\Lambda=K_C+2H$,
-\[
- \tau^2=s(z)\,dz\,\mathfrak f^2,\qquad
- \eta_C=\mathcal E(z)\,dz\,\mathfrak f^2.
-\]
-Let $\ell_q\in W^*$ be coefficient evaluation in $\tau$. Then in the
-dual form on $W^*$,
-\begin{equation}\label{eq:dual-evaluation}
- \ell_q^2=\frac{\mathcal E(q)}{s(q)}\neq 0.
-\end{equation}
-\end{lemma}
-
-\begin{proof}
-The map $f$ is finite and flat of degree four. The identity
-$2\Lambda=K_C+2H$ identifies $\Lambda^2$ with the relative dualizing
-bundle. We check locality and perfection directly.
-
-At a ramification point choose a parameter $w$ with $x=w^e$ on the
-target. A frame of $H$ pulled back from the target can be chosen so
-that $\eta_C=dx\,\mathfrak f^2$. Write $\tau^2=a(w)dw\,\mathfrak f^2$
-with $a$ a unit. The local trace pairing is
-$(b,c)\mapsto\Tr(a(w)bc/(e w^{e-1}))$. On the basis $1,w,\ldots,
-w^{e-1}$ the pairing without the unit $a$ has matrix
-\[
- \Tr\!\left(\frac{w^{i+j}}{e w^{e-1}}\right)
- =\begin{cases}1&i+j=e-1,\\0&i+j\neq e-1,\end{cases}
- \quad 0\le i,j<e.
-\]
-This is holomorphic and perfect; multiplication by the unit $a$ is an
-automorphism. Direct sum over points above a target disk gives
-perfection of the pairing.
-
-On an unramified fiber the trace is the sum of its four contributions,
-with no division by the degree.
-
-The splitting type: $E$ is a vector bundle on $\PP^1$, so a direct sum
-of line bundles. It is self-dual under \eqref{eq:trace-form}, so its
-splitting degrees pair as $(a,b,-b,-a)$ with $a\geq b\geq 0$.
-Proposition~\ref{prop:curve-data} gives
-\[
- h^0(E(-1))=h^0(C,\theta)=3,\qquad
- h^0(E(-2))=h^0(C,H+2Z)=2.
-\]
-For $\OO(\lambda)=E$-summand of degree $\lambda$:
-$h^0(\OO(\lambda-1))=\lambda$ if $\lambda>0$ and $0$ otherwise;
-$h^0(\OO(\lambda-2))=(\lambda-1)$ if $\lambda\ge 2$ and $0$ otherwise.
-The solutions of $a^+ +b^+=3$ and $(a-1)^+(b-1)^+=2$ for
-$a\ge b\ge 0$ give $a=3$, $b=0$. Hence
-$E\simeq\OO(3)\oplus\OO^{\oplus2}\oplus\OO(-3)$.
-
-The map $\OO(3)\to E$ from multiplication by $\xi^2$ has zero
-projection to the $\OO$, $\OO$ and $\OO(-3)$ summands (by degree), so
-its image $\mathcal P$ is a subbundle equal to the positive summand
-and is defined everywhere. Injectivity of $H^0(\PP^1,\OO(3))\to
-H^0(C,3H)$ (both 4-dimensional) gives $H^0(\mathcal P)=\xi^2H^0(C,3H)$.
-
-The positive summand $\mathcal P$ is isotropic because
-$\Hom(\OO(6),\OO)=0$, and it pairs trivially with the two degree-zero
-summands. Perfection gives
-$\mathcal P^\perp\simeq\OO(3)\oplus\OO^{\oplus2}$ and
-$\mathcal P^\perp/\mathcal P\simeq\OO^{\oplus 2}$ with a nondegenerate
-constant form. All global sections of $E$ lie in $\mathcal P^\perp$.
-Taking sections gives $W$ is 2-dimensional and the induced form is
-nondegenerate.
-
-For \eqref{eq:dual-evaluation}, at $q\in Z$ the fiber algebra is
-one-dimensional because $f$ is unramified. The trace pairing restricted
-to the $q$-factor has form coefficient $s(q)/\mathcal E(q)$; evaluation
-at $q$ vanishes on every other factor and on $\mathcal P$. The metric
-dual of coefficient evaluation has, on the factor at $q$, square
-$\mathcal E(q)/s(q)$, and it lies in $\mathcal P_{f(q)}^\perp$.
-Passing to the quotient preserves the square. Finally $\mathcal E(q)
-\neq 0$ by unramifiedness.
-\end{proof}
-
-\subsection{Identification by principal parts}\label{subsec:principal-parts}
-
-The trace form gives a quadratic space on $C$. We now identify it with
-$H^0(\Gamma,\Phi)^*$ via residue duality.
-
-\begin{lemma}\label{lem:principal-parts}
-There is an isomorphism $\iota:W\to H^0(\Gamma,\Phi)$ that identifies
-the full principal parts of $T/\xi^2$ on $C$ and of $\iota(T)/\beta^2$
-on $\Gamma$ via residue duality, and identifies the projective
-evaluations at corresponding points of $Z$. Its dual
-$\iota^*:H^0(\Gamma,\Phi)^*\to W^*$ is an isometry for $Q$
-(Lemma~\ref{lem:aux-quadratic}) and the dual trace form
-(Lemma~\ref{lem:trace-pairing}).
-\end{lemma}
-
-\begin{proof}
-Set $N_\Sigma=\Gamma+F$, $N_\Gamma=F_0+\Delta+F_0+D_0=5F_0+\Delta$
-and $D=2Z$. Fourth-order contact identifies $D$ on $C$ and on $\Gamma$
-as the same embedded length-twelve subscheme. The isomorphism
-$N_\Sigma|_C\simeq K_C-3H$ comes from adjunction against the fixed
-volume (\eqref{eq:adjunction-normalization}). The differential $\omega$
-trivializes $K_\Gamma$.
-
-For a line bundle $B$ on a smooth curve, the principal parts of
-$B(D)$ at $D$ and the jets of $K-B$ are perfect dual by
-\[
- (a,s)\longmapsto\sum_{q\in Z}\Res_q(as).
-\]
-In a parameter and mutually dual frames, coefficients
-$a_{-2}z^{-2}+a_{-1}z^{-1}$ pair with $s_0+s_1z$ as $a_{-2}s_1+a_{-1}
-s_0$. Thus the two full principal-part spaces
-\[
- H^0(C,\OO_C(3H+2Z)/\OO_C(3H)),\qquad
- H^0(\Gamma,\OO_\Gamma(-N_\Gamma+2Z)/\OO_\Gamma(-N_\Gamma))
-\]
-are each identified by residues with
-$H^0(D,N_\Sigma|_D)^*$.
-
-Both complete adjoint systems are restrictions of the same ambient
-system $H^0(\Sigma,N_\Sigma)$. Restriction kernels to $C$ and to
-$\Gamma$ are
-\[
- N_\Sigma-C=(-5,-2),\qquad N_\Sigma-\Gamma=(1,0),
-\]
-with $H^1=0$ by product cohomology (both indices nonnegative or
-appropriate for $H^1=0$). Their images on the common $D$ therefore
-coincide. Equivalently, an ambient section vanishing on $\Gamma$
-vanishes to order at least four on $C$ at $Z$ and contributes nothing
-to these residue tests.
-
-The principal-part exact sequence and Serre duality say the image of
-$H^0(C,\Lambda)$ by $T\mapsto\operatorname{pp}(T/\xi^2)$ is precisely
-the annihilator of the common adjoint jet space, and its kernel is
-$\xi^2H^0(C,3H)$. On $\Gamma$ the same is true for
-\[
- H^0(\Gamma,\Phi)\longrightarrow
- H^0(\Gamma,\OO_\Gamma(-N_\Gamma+2Z)/\OO_\Gamma(-N_\Gamma)),
- \quad \widetilde T\longmapsto\operatorname{pp}(\widetilde T/\beta^2),
-\]
-since $\Phi=2M-N_\Gamma$ and $(\beta)=Z$. This map is injective since
-its kernel would be a section of $-N_\Gamma$ (degree $-10$). The two
-identifications with the same annihilator yield
-$\iota:W\to H^0(\Gamma,\Phi)$, identifying full principal parts. As a
-sanity check, $h^0(N_\Gamma)=\deg N_\Gamma=10$ and restriction to $D$
-is injective since $\deg(N_\Gamma-2Z)=-2$, so the annihilator of the
-image has $12-10=2$ dimensions.
-
-For the metric, near $q\in Z$ choose $(y,z)$ and frames $\mathfrak g$
-of $\Gamma$ and $\mathfrak f$ of $F$ with $\gamma=y\mathfrak g$,
-$\Omega=e(y,z)(dy/y)\wedge dz$, $e(0,z)=1$. Write
-\[
- \kappa=k(y,z)\mathfrak g^2\mathfrak f^4,\qquad
- \beta=b(z)\mathfrak m,\qquad
- \mathfrak m^4=\mathfrak g^2\mathfrak f^4|_\Gamma,
-\]
-so that $k(0,z)=b(z)^4$ and $b'(0)k_y(0,0)\neq 0$. Put
-$K(z)=k_y(0,z)$; then the equation of $C$ reads
-\[
- y|_C=-\frac{b(z)^4}{K(z)}+O(z^8),\quad
- e|_C=1+O(z^4),\quad k_y|_C=K(z)+O(z^4).
-\]
-The adjunction convention sends the frame
-$\mathfrak g\mathfrak f^4$ to
-\begin{equation}\label{eq:adjunction-frame}
- \Res_C\!\left(\frac{e(y,z)dy\wedge dz}{k(y,z)}\right)
- =\frac{e|_C}{k_y|_C}\,dz.
-\end{equation}
-Sign is positive because $dy\wedge dz=(dk\wedge dz)/k_y$.
-
-Let $a=\xi/z$ be the local frame of $\OO_C(Z)$ and $\tau=\mathfrak
-f^3a^2$ the resulting frame of $\Lambda$. The fourth-power
-identification sends $a^4$ to $(y|_C/z^4)\mathfrak g$, so by
-\eqref{eq:adjunction-frame},
-\[
- \tau^2=s(z)\,dz\,\mathfrak f^2,\qquad
- s(z)=\frac{e|_C y|_C}{z^4 k_y|_C},\qquad
- s(0)=-\frac{b'(0)^4}{k_y(0,0)^2}.
-\]
-The minus sign comes from solving for $y|_C$.
-
-Write $T=\pi(z)\tau$ and $\iota(T)=\widetilde T=\widetilde\pi(z)
-\widetilde\tau$ in the frame
-$\widetilde\tau=\mathfrak m^2/(\mathfrak g\mathfrak f)$ of $\Phi$.
-For an ambient test section $A(y,z)\mathfrak g\mathfrak f$ the two
-residue tests are
-\[
- \Res_0\!\left(\frac{A|_C\,\pi\,e|_C}{k_y|_C\,z^2}\,dz\right),\qquad
- \Res_0\!\left(\frac{A(0,z)\widetilde\pi}{b(z)^2}\,dz\right).
-\]
-The restrictions of $A$ to the two curves agree modulo $z^4$; the
-equality of the functionals for all test jets is exactly
-\[
- \operatorname{pp}\!\left(\frac{\pi}{K(z)z^2}\right)
- =\operatorname{pp}\!\left(\frac{\widetilde\pi}{b(z)^2}\right),
- \qquad
- \widetilde\pi\equiv\frac{(b(z)/z)^2}{K(z)}\,\pi\pmod{z^2}.
-\]
-Writing $b=b_1z+b_2z^2+\cdots$, $K=K_0+K_1z+\cdots$,
-$\pi=\pi_0+\pi_1z+\cdots$, both principal part functionals yield the
-two-coefficient triangular transformation
-\[
- \widetilde\pi_0=\frac{b_1^2}{K_0}\pi_0,\qquad
- \widetilde\pi_1=\frac{b_1^2}{K_0}\pi_1
- +\left(\frac{2b_1b_2}{K_0}-\frac{b_1^2K_1}{K_0^2}\right)\pi_0.
-\]
-This is invertible and coordinate-independent.
-
-For the metric, write $\eta_C=\mathcal E_C\,dz\,\mathfrak f^2$ and
-$\eta_\Gamma=\mathcal E_\Gamma\,dz\,\mathfrak f^2$. Tangency gives
-$\mathcal E_C(0)=\mathcal E_\Gamma(0)$. If $\ell$ and $\widetilde\ell$
-are coefficient evaluations in $\tau$ and $\widetilde\tau$, then
-$\iota^*\widetilde\ell=\alpha\ell$ with $\alpha=b_1^2/K_0$. By
-\eqref{eq:dual-evaluation} and the formula for $s(0)$,
-\[
- (\iota^*\widetilde\ell)^2
- =\alpha^2\frac{\mathcal E_C(0)}{s(0)}
- =-\mathcal E_\Gamma(0).
-\]
-The compatible bundle identification gives $\widetilde\tau^2=
-\mathfrak f^2$, so the last quantity is exactly the value assigned by
-$Q$ to $\widetilde\ell\in H^0(\Gamma,\Phi)^*$, since the section
-defining $Q$ multiplies to $-\eta_\Gamma/\omega$. The evaluations at
-$Z$ have three distinct projective directions; a binary quadratic
-that matches $Q$ at all three is identically equal. Hence $\iota^*$
-is an isometry, with sign and scalar fixed.
-\end{proof}
-
-\subsection{The meromorphic section with prescribed residues}\label{subsec:residue-data}
-
-We now convert the abstract isometry into concrete pointwise data at
-the six marked points.
-
-On a disk centered at $q_j$ choose a parameter $z_j$ and a frame
-$\ell_j^{\mathrm{fr}}$ of $L$ whose fourth power maps to $dz_j$ under
-the fixed isomorphism $L^4=K_C$. Set
-\[
- a_j=\xi/z_j,\quad
- \mathfrak f_j=\ell_j^{\mathrm{fr}}/a_j,\quad
- \tau_j=(\ell_j^{\mathrm{fr}})^2\mathfrak f_j;
-\]
-these frame $\OO_C(Z)$, $H$ and $\Lambda$ respectively. If
-$r=\xi u$, then $r=z_j u$ in these frames. The fiber-linear function
-$t_j$ on $L^{-1}$ dual to $\ell_j^{\mathrm{fr}}$ gives compatible
-split coordinates $(t_j,z_j)$.
-
-Since $\tau_j^2=dz_j\,\mathfrak f_j^2$, Lemma~\ref{lem:trace-pairing}
-says that if $\ell_j\in W^*$ denotes coefficient evaluation in
-$\tau_j$, then
-\begin{equation}\label{eq:compatible-norm}
- \ell_j^2=(u_1u_2'-u_2u_1')(0).
-\end{equation}
-Here and below primes denote $z_j$-derivatives.
-
-\begin{proposition}[Residue data]\label{prop:residue-data}
-After possibly a determinant-one change of basis $u$ (preserving
-$Q$, $L$, and the normalization), with the zero divisor of $u_1$ a
-reduced fiber of $f$ disjoint from $Z$, the following hold. Set
-$r=\xi u$, $v_j^*=\iota^*w_j\in W^*$ and $V=\sum_{j=1}^6 v_j^*$. For
-any mutually disjoint compatible coordinate disks as above, there are
-nonzero constants $c_j\in\CC$ and a pair
-$P\in H^0(C,\theta+Z)^{\oplus 2}$ such that:
-\begin{enumerate}
-\item $c_j\ell_j=v_j^*$ for each $j$;
- $V^2=90=9\deg L$; and $(v_j^*)^2+v_j^*\cdot V=1$ for each $j$.
-\item On disk $j$, in the compatible frames, put
- $m_j=c_j/z_j$, $R_j=m_jr$, $W_j=R_j(0)\wedge R_j'(0)$, and
- $p_j=P-m_jr'$. Then $P$ has principal part $R_j(0)/z_j$
- in the frame $(\ell_j^{\mathrm{fr}})^2$ of $\theta$; the rows
- $R_j,p_j$ are holomorphic near $0$; $R_j(0)\neq 0$.
-\item Residue identities at each $j$:
- \[
- W_j=(v_j^*)^2,\qquad p_j(0)\wedge R_j(0)=1,\qquad
- \sum_{j=1}^6(W_j-1)=-9\deg L.
- \]
-\item Global residue pairing. With $J=u\wedge P\in H^0(C,\Lambda)$,
- \[
- \langle[J],[T]\rangle_W=-V([T])
- \quad\text{for every }T\in H^0(C,\Lambda).
- \]
-\item Changing $P$ by a holomorphic pair of sections of $\theta$ does
- not change $[J]$. The pole prescription is invariant under changes
- of compatible coordinates.
-\end{enumerate}
-\end{proposition}
-
-\begin{proof}
-A determinant-one change of basis of $u$ preserves $\eta_C$ and
-$\eta_\Gamma$ and hence the form $Q$. Avoiding the finitely many
-branch values and the image of $Z$, we can make $u_1$'s zero divisor
-a suitable reduced fiber. The isometry $\iota^*$ transports the
-equations for $w_j$ to
-$c_j\ell_j=v_j^*$; since $\ell_j$ is nonzero and $v_j^*\neq 0$ there
-is a unique such $c_j$. Locally $R_j=c_ju$ is holomorphic and nonzero
-at the origin, and \eqref{eq:compatible-norm} gives
-$R_j(0)\wedge R_j'(0)=c_j^2\ell_j^2=(v_j^*)^2$.
-
-Every prescribed set of simple principal parts of $\theta$ at $Z$ is
-realisable by a section of $\theta+Z$: the obstruction lies in
-$H^1(C,\theta)$, which by Serre duality pairs with
-$H^0(C,K_C-\theta)=H^0(C,\theta)$. Every section of $\theta$ vanishes
-twice along $Z$ (Proposition~\ref{prop:curve-data}(3)), so all such
-residues vanish. Applying this separately to both entries produces
-$P$ with the prescribed principal part. Then $m_jr'$ has the same
-simple pole part, so $p_j=P-m_jr'$ is holomorphic.
-
-The simple pole of $P$ at $q_j$ is proportional to $u(q_j)$; hence
-$u\wedge P$ is holomorphic there. Away from $Z$ it is already
-holomorphic. So $J=u\wedge P\in H^0(C,\Lambda)$. To compute its
-class, fix $T\in H^0(C,\Lambda)$ and consider the meromorphic
-canonical differential $\nu=P_1T/u_1$; its degree is
-$\theta+(\theta+H)-H=2\theta=K_C$. Its only possible poles are at $Z$
-and at the simple zero fiber of $u_1$. At $q_j$ the compatible frames
-make the canonical product equal to the coefficient product times
-$dz_j$, so $\Res_{q_j}\nu=c_j\ell_j(T)$. At a zero $q$ of $u_1$, the
-section $u_2$ is nonzero, $J(q)=-u_2(q)P_1(q)$ and
-$\eta_C(q)=-u_2(q)du_1(q)$, so the residue is
-$(JT/\eta_C)(q)$. Summing residues on the zero fiber gives the value
-of $\Tr(JT/\eta_C)$ there; this trace is a global holomorphic function
-on $\PP^1$ (Lemma~\ref{lem:trace-pairing}), hence constant. The
-residue theorem gives
-\[
- \langle[J],[T]\rangle_W
- =-\sum_j c_j\ell_j(T)=-V([T]),
-\]
-which is the stated sign.
-
-Locally, since $r=zu$,
-\[
- P=p+\frac{c}{z}u+cu',\qquad cJ=R\wedge p+R\wedge R'.
-\]
-So $c_j\ell_j(J)=R_j(0)\wedge p_j(0)+W_j$ on disk $j$. On the other
-hand, \eqref{eq:trace-form} applied to $\iota^*$ gives
-\[
- c_j\ell_j(J)=-v_j^*\cdot V=W_j-1,
-\]
-because $v_j^*\cdot V=1-W_j$. Thus $R_j(0)\wedge p_j(0)=-1$, i.e.\
-$p_j(0)\wedge R_j(0)=1$. Summing $W_j-1=-v_j^*\cdot V=-v_j^*\cdot V$
-over $j$ yields $\sum_j(W_j-1)=-V\cdot V=-V^2=-90=-9\deg L$.
-
-The useful local expansion, used below, is
-\begin{equation}\label{eq:P-local-expansion}
- P=\frac{R_j(0)}{z_j}+p_j(0)+2R_j'(0)+O(z_j).
-\end{equation}
-If $P$ is replaced by $P+S$ with $S\in H^0(C,\theta)^{\oplus 2}$, then
-$u\wedge S\in\xi^2H^0(C,3H)$ by Proposition~\ref{prop:curve-data}(3),
-so $[J]$ is unchanged.
-
-Finally, invariance under compatible coordinate changes: let
-$\zeta=\psi(z)$ be another centered parameter and $g(z)\ell^
-{\mathrm{fr}}$ the corresponding frame, so $g^4=\psi'$. The frame
-transforms have
-\[
- a_{\rm new}=\frac{z}{\psi(z)}a,\qquad
- \mathfrak f_{\rm new}=g\frac{\psi(z)}{z}\mathfrak f,\qquad
- \tau_{\rm new}=g^3\frac{\psi(z)}{z}\tau.
-\]
-Writing $g_0=g(0)$, one gets $\ell_{\rm new}=g_0^{-7}\ell$, so the
-same $v^*$ prescribes $c_{\rm new}=g_0^7c$. Also $r_{\rm new}=g^{-1}r$
-and $R_{\rm new}(0)=g_0^2R(0)$. The coefficient row of the global
-$\theta$-section transforms as $P_{\rm new}=g^{-2}P$; its
-$\zeta^{-1}$-coefficient is $\psi'(0)g_0^{-2}R(0)=g_0^2R(0)$, exactly
-the new prescribed $R_{\rm new}(0)$, so the same global $P$ realizes
-the new prescription. Differentiating $R_{\rm new}=g_0^2R$ at $0$ in
-the new coordinate gives $W_{\rm new}=W$; and
-$p_{\rm new}(0)-g_0^{-2}p(0)$ is proportional to $R(0)$, so
-$p_{\rm new}(0)\wedge R_{\rm new}(0)=p(0)\wedge R(0)$.
-\end{proof}
-
-\section{Gluing the vector-field jets}\label{sec:jets}
-
-We now do the actual construction on the total space of $L^{-1}$. The
-task is to pick finite-order jets of two vector fields whose determinant
-is a unit times a prescribed factor, so that the next section can lift
-them holomorphically. This is the only place where substantial
-bookkeeping happens.
-
-\subsection{Coordinates, transitions, and obstruction sheaves}\label{subsec:coords}
-
-On the total space of $L^{-1}$ choose a base coordinate $z$ and a
-fiber-linear coordinate $t$ measuring the equation of the zero section.
-We work with the local frame $(E_t,\partial_z)$ where $E_t=t\partial_t$.
-Choose coordinates adapted to $L^4=K_C$: under $z\mapsto\widetilde z
-=\psi(z)$ we require $\widetilde t=g(z)t$ with $g^4=\psi'$. The
-meromorphic two-form
-\begin{equation}\label{eq:volume-form}
- \sigma=t^{-5}\,dt\wedge dz
-\end{equation}
-is invariant under such split changes. We seek rows
-$A=(A_1,A_2)$, $B=(B_1,B_2)$ of functions of $(t,z)$ defining
-\begin{equation}\label{eq:vector-fields}
- v_i=t(A_i(t,z)E_t+B_i(t,z)\partial_z)\quad(i=1,2),
-\end{equation}
-with Taylor expansions $A=\sum_{k\ge 0}A_k t^k$,
-$B=\sum_{k\ge 0}B_k t^k$ where $A_k=(A_{k1},A_{k2})$ and
-$B_k=(B_{k1},B_{k2})$. At leading order,
-\begin{equation}\label{eq:leading-coefficients}
- A_0=r,\qquad B_0=0.
-\end{equation}
-Then
-\begin{equation}\label{eq:determinant-form}
- \sigma(v_1,v_2)=t^{-2}(A\wedge B),\qquad
- v_1\wedge v_2=t^2(A\wedge B)E_t\wedge\partial_z.
-\end{equation}
-
-Under a split change $(\widetilde z,\widetilde t)=(\psi(z),g(z)t)$
-of the above type, put $a=(\log g)'$. A term
-$t^{k+1}(A_kE_t+B_k\partial_z)$ transforms by
-\begin{equation}\label{eq:split-transform}
- \widetilde A_k=g^{-(k+1)}(A_k+aB_k),\qquad
- \widetilde B_k=g^{3-k}B_k.
-\end{equation}
-Thus $A_k$ has \emph{type} $L^{k+1}$ (scales by $g^{-(k+1)}$) and
-$B_k$ has \emph{type} $L^{k-3}$ (scales by $g^{3-k}$). The
-homogeneous-term sheaf at order $k$ fits into the extension
-\begin{equation}\label{eq:homogeneous-extension}
- 0\to L^{k+1}\to\mathcal E_k\to L^{k-3}\to 0.
-\end{equation}
-
-For the cover: take $U_0=C\setminus Z$ and six mutually disjoint disks
-$U_j$ about the $q_j$. A positively large multiple of $Z$ gives an
-ample line bundle; the complement of a general hyperplane section in
-its very ample embedding is affine, hence $U_0$'s analytification is
-Stein. The disks and annular intersections are Stein as well. By
-\eqref{eq:homogeneous-extension} and Cartan's Theorem B on Stein
-spaces~\cite{Cartan1952}, horizontal sections admit holomorphic
-vertical lifts on each chart. Obstruction computations below use
-contour residues $\frac{1}{2\pi i}\int\omega$ around omitted points,
-which is a form of Serre duality for overlap functions holomorphic on
-annuli; see~\cite{SerreDuality1955}.
-
-\subsection{Transition generators and the bracket}\label{subsec:generators}
-
-The changes of gluing are compositions of time-one flows of holomorphic
-vector fields
-\begin{equation}\label{eq:generator}
- D=t^k(bE_t+h\partial_z),\qquad k>0,
-\end{equation}
-where $b,h$ are holomorphic on the annular overlap in the disk
-coordinate. Use the flow of $-D$, so that the action on other vector
-fields is $\exp(\mathrm{ad}\,D)$. Each generator vanishes on the zero
-section and changes its equation only by $t\mapsto t+O(t^2)$, so the
-zero section and its conormal bundle are preserved.
-
-\begin{lemma}\label{lem:bracket}
-For $D$ as in \eqref{eq:generator} and
-$V=t^{\ell+1}(AE_t+B\partial_z)$,
-\begin{align}
- [D,V]=t^{k+\ell+1}\bigl(&((\ell+1-k)bA+hA'-Bb')E_t\notag\\
- &+((\ell+1)bB+hB'-khA-Bh')\partial_z\bigr).
- \label{eq:general-bracket}
-\end{align}
-Moreover, the logarithmic divergence
-\begin{equation}\label{eq:divergence}
- \mathcal L_D\sigma=t^k\bigl((k-4)b+h'\bigr)\sigma.
-\end{equation}
-For $k\neq 4$ the choice $b=h'/(4-k)$ preserves $\sigma$ exactly.
-The first effect on the leading row $(A,B)$ of~\eqref{eq:vector-fields}
-is
-\begin{equation}\label{eq:leading-action}
- \Delta A_k=(1-k)br+hr',\qquad
- \Delta B_k=-khr.
-\end{equation}
-\end{lemma}
-
-\begin{proof}
-Compute $[t^k bE_t,t^{\ell+1}AE_t]$ etc.\ in the two components of
-the bracket. For the divergence, the coefficient of $dt\wedge dz$ in
-$\sigma$ is $t^{-5}$, so
-\[
- \mathrm{div}_\sigma D
- =t^5\partial_t(t^{k-4}b)+t^5\partial_z(t^{k-5}h)
- =t^k((k-4)b+h').
-\]
-Setting $\ell=0$, $A=r$, $B=0$ in the bracket gives
-\eqref{eq:leading-action}.
-\end{proof}
-
-On a compact subannulus every $D$ in \eqref{eq:generator} is
-holomorphic near the zero section and vanishes there; so its time-one
-flow and inverse exist after shrinking the fiber disc. A finite
-composition has the computed Taylor series. There are no triple
-overlaps, so no additional cocycle relation is imposed. The resulting
-transition germs will be realized on an actual surface in
-Section~\ref{sec:analytic}.
-
-\subsection{The first two orders}\label{subsec:first-two}
-
-Order one. On the $j$-th overlap, take $h=m=m_j$ and $b=m'/3$. Choose
-the inside and outside values
-\begin{equation}\label{eq:order-one}
-\begin{array}{c|cc}
- &A_1&B_1\\\hline
- U_j&p&mr\\
- U_0&P&0
-\end{array}
-\end{equation}
-By \eqref{eq:leading-action} the first transition adds
-$(mr',-mr)$ to the inside row, giving exactly $(P,0)$ on the outside.
-The degree-one coefficient of $A\wedge B$ vanishes on both sides.
-
-Order two. Write $U=B_2$. On each chart solve
-\begin{equation}\label{eq:order-two-horizontal}
- r\wedge U=1-A_1\wedge B_1.
-\end{equation}
-On the disk write $r=z\rho$ with $\rho$ nowhere zero after shrinking.
-The right-hand side vanishes at $z=0$ by \eqref{eq:P-local-expansion}
-in Proposition~\ref{prop:residue-data}, so division by $z$ reduces
-\eqref{eq:order-two-horizontal} to a surjective wedge with $\rho$.
-On $U_0$, $r$ has no common zero and the corresponding wedge map of
-coherent sheaves is surjective on sections by Steinness. So holomorphic
-$U$ exists. After the order-one transition, the difference of the two
-chosen $U$'s on an overlap wedges to zero with $r$; it is therefore a
-unique holomorphic multiple of $r$. An order-two transition with
-$b=h'/2$ removes it (its horizontal effect is $-2hr$). Fix the
-outside $U$ throughout this matching.
-
-It remains to match vertical lifts $A_2$. The obstruction lies in
-$H^1(C,L^3)^2$, tested against $H^0(C,K_C\otimes L^{-3})=H^0(C,L)$
-with the two components $r_1,r_2$ as Serre-dual sections.
-
-\begin{lemma}\label{lem:order-two-obstruction}
-The order-two vertical obstruction vanishes for all four entries.
-\end{lemma}
-
-\begin{proof}
-Write $D^{(2)}=A_2$ as the vertical row (not a transition generator).
-For $i,j\in\{1,2\}$ put
-\begin{equation}\label{eq:psi-order-two}
- \Psi_{ij}
- =\left(r_iD^{(2)}_j+\tfrac14 r_i'U_j+\tfrac34 r_j'U_i\right)dz.
-\end{equation}
-Once horizontal rows match, the difference of $\Psi_{ij}$ for inside
-and outside is exactly $r_i$ times the $j$-th vertical difference,
-multiplied by $dz$.
-
-Put $\epsilon_{12}=1$, $\epsilon_{21}=-1$, $\epsilon_{11}=\epsilon_{22}
-=0$. Under a split change with $a=(\log g)'$,
-\[
- \widetilde r_i=g^{-1}r_i,\quad
- \partial_{\widetilde z}\widetilde r_i=g^{-5}(r_i'-ar_i),\quad
- \widetilde U_i=gU_i,\quad
- \widetilde D^{(2)}_i=g^{-3}(D^{(2)}_i+aU_i).
-\]
-Substituting in \eqref{eq:psi-order-two},
-\[
- \widetilde\Psi_{ij}-\Psi_{ij}
- =\tfrac34\epsilon_{ij}\,d\log g
- \quad\text{on }U_0,
-\]
-using $r\wedge U=1$ on the outside. The section $r_1=\xi u_1$ of $L$
-has $\widetilde r_1=g^{-1}r_1$, so
-$\Psi_{ij}+\tfrac34\epsilon_{ij}d\log r_1$ is intrinsic on $U_0$ and
-has only simple poles at the four zeros of $u_1$, each of residue
-$\tfrac34\epsilon_{ij}$. The residue theorem on the complement of
-small disks gives total contour residue $-3\epsilon_{ij}$ for this
-intrinsic differential, and subtracting the six simple logarithmic
-residues at $Z$ yields
-\begin{equation}\label{eq:outside-total}
- \sum_{q\in Z}\Res_q\Psi_{ij}^{\mathrm{out}}
- =-\tfrac34\epsilon_{ij}\deg L.
-\end{equation}
-This argument uses only contour integrals at $Z$ and makes no
-meromorphic extension assumption.
-
-Inside. Before transport, $\Psi_{ij}$ is holomorphic on the disk.
-The order-two transition changes
-\[
- \delta D^{(2)}=hr'-\tfrac12h'r,\qquad\delta U=-2hr,
-\]
-so $\delta\Psi_{ij}=-\tfrac12d(hr_ir_j)$, an exact differential with
-zero contour residue. It remains to compute the effect of the order-one
-transition, including its second exponential term.
-
-The bracket of $t(m'E_t/3+m\partial_z)$ with $t^2(pE_t+mr\partial_z)$
-gives
-\[
- \left(\tfrac13m'p+mp'-\tfrac13mm''r,\quad
- \tfrac23mm'r+m^2r'-mp\right)
-\]
-in $(A_2,U)$, and half of the second bracket with $trE_t$ gives
-\[
- \left(\tfrac23mm'r'+\tfrac12m^2r''+\tfrac16mm''r,\quad
- -\tfrac13mm'r-m^2r'\right).
-\]
-Summing:
-\begin{align}
- \Delta A_2&=\tfrac13m'p+mp'+\tfrac23mm'r'
- +\tfrac12m^2r''-\tfrac16mm''r,\notag\\
- \Delta U&=-mp+\tfrac13mm'r.
- \label{eq:second-order-expansion}
-\end{align}
-Abbreviate $R=mr$, $m=c/z$, so $r=zR/c$. Substituting in
-\eqref{eq:second-order-expansion}:
-\[
- \Delta A_2=-\frac{cR}{z^3}+\frac{cR'}{3z^2}
- +\frac{cR''}{2z}-\frac{cp}{3z^2}+\frac{cp'}{z},\qquad
- \Delta U=-\frac{cp}{z}-\frac{cR}{3z^2}.
-\]
-Equating the Laurent residues of $r_i\Delta A_{2,j}$ and
-$r_i'\Delta U_j$ with coefficients at $0$:
-\begin{align}
- F_{ij}=\Res_0(r_i\Delta A_{2,j}\,dz)
- &=-\tfrac13R_ip_j-\tfrac23R_iR_j'-R_i'R_j,\notag\\
- H_{ij}=\Res_0(r_i'\Delta U_j\,dz)
- &=-R_ip_j-\tfrac13R_iR_j'-\tfrac23R_i'R_j.
- \label{eq:laurent-residues}
-\end{align}
-The total inside residue is thus
-\begin{equation}\label{eq:total-residue-matrix}
- T_{ij}=F_{ij}+\tfrac14H_{ij}+\tfrac34H_{ji}
- =-\tfrac7{12}R_ip_j-\tfrac9{12}R_jp_i
- -\tfrac54R_iR_j'-\tfrac{17}{12}R_i'R_j.
-\end{equation}
-Decompose $T$ into symmetric and antisymmetric parts. The symmetric
-part is
-\[
- T_{ij}^{\mathrm{sym}}
- =-\tfrac23(R_ip_j+R_jp_i+2R_iR_j'+2R_i'R_j),
-\]
-and the antisymmetric part is
-\[
- T_{ij}^{\mathrm{alt}}
- =\tfrac1{12}\epsilon_{ij}
- \bigl((R\wedge p)(0)+(R\wedge R')(0)\bigr).
-\]
-To identify the symmetric part intrinsically, observe that
-$P=p+mr'=\frac{R(0)}{z}+p(0)+2R'(0)+O(z)$ and $P_iP_j$ is a
-meromorphic section of $L^4=K_C$ with local expression $P_iP_j\,dz$.
-Then $\Res_0(P_iP_j\,dz)=R_ip_j+R_jp_i+2R_iR_j'+2R_i'R_j$, so the sum
-of every symmetric entry over $Z$ vanishes by the residue theorem on
-$C$. For the antisymmetric part, $R(0)\wedge p(0)=-1$ and
-\[
- \sum_{q\in Z}T_{ij}^{\mathrm{alt}}
- =\tfrac1{12}\epsilon_{ij}\sum_{\nu=1}^6(W_\nu-1)
- =-\tfrac34\epsilon_{ij}\deg L.
-\]
-So the total inside sums are
-$(0,-\tfrac34\deg L\cdot\epsilon_{12},\tfrac34\deg L\cdot
-\epsilon_{21},0)$, which equal the four outside totals in
-\eqref{eq:outside-total}. Every Serre-duality pairing of the vertical
-difference vanishes, so the corresponding class in $H^1(C,L^3)^2$ is
-zero and holomorphic vertical corrections complete the order-two
-matching.
-\end{proof}
-
-\subsection{Higher orders and determinant targets}\label{subsec:higher-orders}
-
-From order three on, fields and determinant are extended together.
-Write the determinant equation as
-\begin{equation}\label{eq:target}
- A\wedge B=t^2G,\qquad G=1+G_1t+G_2t^2+\cdots.
-\end{equation}
-The field equation in degree $k$ uses $G_{k-2}$; to compare horizontal
-rows we need target compatibility already established. The target is
-the local bivector $G\sigma^{-1}$, living in the line bundle
-\[
- \mathcal M=\cI^4\det\cF=\cI^5\det\cT_Y
-\]
-with $\mathcal M|_C=L^4\otimes\cT_C\simeq\OO_C$ (using $L^4=K_C$).
-The local frame of $\mathcal M$ is $\sigma^{-1}$; the degree-$k$
-associated graded is $L^k$. Once transitions stop preserving $\sigma$,
-$G$ is not an invariant scalar. The transformation law is: if
-$q=\mathrm{div}_\sigma D$, then
-\begin{equation}\label{eq:target-transform}
- [D,G\sigma^{-1}]=(D(G)-qG)\sigma^{-1}.
-\end{equation}
-An order-$k$ transition changes a target with constant term $1$ first
-in degree $k$ of $G$, with contribution $-((k-4)b+h')$.
-
-The factor $t^2$ in \eqref{eq:target} and $B_0=0$ determine the order
-in which the choices can be made: the field coefficient $A_a$ appears
-in $A\wedge B$ at degrees $\ge a+1$, and $B_a$ at degrees $\ge a$.
-Concretely for step $k$:
-\begin{itemize}
-\item a global addition to $A_{k-1}$ affects degree $k$ of $A\wedge B$
- (i.e.\ $G_{k-2}$);
-\item $B_k$ affects degree $k$ (also $G_{k-2}$);
-\item $A_k$ affects degree $k+1$ (i.e.\ $G_{k-1}$).
-\end{itemize}
-All targets referenced here are already chosen and fixed. So we can
-freely change $A_{k-1}$ to solve the current equation without breaking
-earlier ones, and $G_k$ will be picked only after fields match through
-index $k$.
-
-We maintain the following invariant after step $n\geq 2$: field rows
-through index $n$ match; determinant equations are imposed through
-degree $n$ in $A\wedge B$; coefficients $G_0=1,\ldots,G_n$ are chosen
-and transform compatibly. This is true for $n=2$ with $G=1$ because
-both transitions of Section~\ref{subsec:first-two} preserve $\sigma$
-exactly.
-
-Assume it holds through $k-1$ for $k\ge 3$. On each chart we need
-\begin{equation}\label{eq:horizontal-induction}
- r\wedge B_k=G_{k-2}-\sum_{a=1}^{k-1}A_a\wedge B_{k-a}.
-\end{equation}
-This is an equation in $L^{k-2}$. Indeed, under
-\eqref{eq:split-transform} the extra terms on the right are multiples
-of $\sum_{a=1}^{k-1}B_a\wedge B_{k-a}=0$.
-
-\paragraph{Making \eqref{eq:horizontal-induction} divisible at $Z$.}
-Add a row $a\in H^0(C,L^k)^2$ to $A_{k-1}$. Such a change preserves
-matching through index $k-1$ and all determinant equations already
-imposed (because $B_0=0$: the new term $t^kaE_t$ first contributes
-$a\wedge B_1$ in degree $k$, and the equation for degree $k$ has not
-yet been imposed). At each $q_j$, the row $B_1(q_j)=R_j(0)$ is
-nonzero, and \eqref{eq:evaluation-surjectivity} (
-Proposition~\ref{prop:curve-data}(4)) gives
-$H^0(C,L^k)\to H^0(Z,L^k|_Z)$ surjective for $k\ge 3$. The scalar
-maps $a(q_j)\mapsto a(q_j)\wedge B_1(q_j)$ are surjective, so we can
-make the right side of \eqref{eq:horizontal-induction} vanish at each
-$q_j$. Since $r=z\rho$ has only a simple common zero, no derivative
-condition is needed. Dividing by $z$ on each disk and using Steinness
-on $U_0$ gives holomorphic solutions $B_k$.
-
-The previous transitions act on $a$ in the current degree. Only the
-first transition contributes at row index $k$:
-\begin{equation}\label{eq:old-transition-effect}
- \delta A_k=\tfrac{k-1}{3}m'a+ma',\qquad\delta B_k=-ma.
-\end{equation}
-All later generators and repeated brackets affect strictly higher
-indices. The horizontal contribution satisfies $r\wedge(-ma)=a\wedge
-(mr)$, exactly the corresponding determinant change on the disk. We
-incorporate these in the matching below.
-
-\paragraph{Matching horizontal rows.}
-Choose holomorphic vertical lifts of the new $B_k$. After transport
-by previous transitions, lower degree-$k$ rows already agree. The
-degree-$k$ difference of determinants is $r\wedge\delta B_k$ (because
-$\delta A_k\wedge B_0=0$). Target compatibility through degree $k-2$
-of $G$ makes this difference zero. So the horizontal discrepancy is a
-holomorphic multiple of $r$ on each overlap, and \eqref{eq:leading-action}
-removes it by a choice of $h$ in an order-$k$ transition. Use $b=h'$
-when $k=3$ and $b=0$ when $k\ge 4$. The new transition first affects
-$G$ in degree $k$, so it does not alter the target comparison in
-degree $k-2$ just used.
-
-\paragraph{Matching vertical rows.}
-After horizontal matching, the remaining class lies in
-$H^1(C,L^{k+1})^2$.
-For $k\ge 4$ it vanishes because
-\begin{equation}\label{eq:vertical-vanishing}
- H^1(C,L^{k+1})\simeq H^0(C,L^{3-k})^*=0.
-\end{equation}
-Holomorphic vertical corrections on the charts solve the matching.
-
-For $k=3$ the obstruction lies in $H^1(C,K_C)^2$, tested by the
-constants in the two entries. Additional overlap freedom kills it: on
-the $j$-th overlap, replace $h$ by $h+\lambda_j/z$ and compensate by
-$B_3\mapsto B_3+3\lambda_jr/z$. This is holomorphic, and preserves
-both the current determinant equation and horizontal matching (the
-transition adds horizontal row $-3\lambda_jr/z$, its negation). Keep
-$A_3$ unchanged in the disk coordinate. The change in the transported
-vertical difference at index three is
-\begin{equation}\label{eq:order-three-residue}
- \lambda_j\!\left(\frac{r'}z+\frac{2r}{z^2}\right),\qquad
- \Res_0\!\left[\lambda_j\!\left(\frac{r'}z+\frac{2r}{z^2}\right)dz\right]
- =3\lambda_jr'(0).
-\end{equation}
-The six rows $r'(q_j)$ span $\CC^2$ (Section~\ref{sec:curve}), so
-$(\lambda_1,\ldots,\lambda_6)\mapsto3\sum_j\lambda_jr'(q_j)$ has
-rank two and cancels both obstruction entries. Order three is complete.
-The correction \eqref{eq:order-three-residue} can change the degree-four
-coefficient of $A\wedge B$; that is harmless because $G_2=0$ has been
-chosen but its determinant equation is first imposed at step four.
-The modified order-three transition still has $b=h'$ and preserves
-$\sigma$ exactly.
-
-\paragraph{Extending the determinant target.}
-Choose $G_k$ compatibly after the order-$k$ transition.
-For $k=3$, all transitions preserve $\sigma$, so $G=1$.
-For $k=4$: the first target discrepancy from the order-four transition
-with $b=0$ is $-h'$ in degree four. The horizontal coefficient of an
-order-four generator has type $L^4\otimes\cT_C=\OO_C$; more explicitly
-$\widetilde h=g^{4-k}h=h$, so $h'dz=dh$ is an intrinsic exact
-differential on the overlap. Its contour residue vanishes; its
-pairing with the constant section of $\OO_C$ (the only Serre-dual test)
-also vanishes because the residue is zero. So the cocycle splits and
-gives holomorphic chart functions $G_4$ with $G_1=G_2=G_3=0$.
-For $k>4$, once lower coefficients are compatible, the new leading
-discrepancy lies in $L^k$, and
-\begin{equation}\label{eq:target-vanishing}
- H^1(C,L^k)\simeq H^0(C,L^{4-k})^*=0,
-\end{equation}
-so it always splits and defines $G_k$. In every case the extension
-leaves all earlier equations alone (since $G_k$ is used only in degree
-$k+2$ of $A\wedge B$). This completes the induction invariant.
-
-We stop at index seven. A row of index $k$ contributes degree $k+1$ in
-the logarithmic frame $(E_t,\partial_z)$, so matching through index
-seven prescribes a section modulo $\cI^9\cF$. This threshold is
-needed for the vanishing argument of Section~\ref{sec:analytic}: the
-unprescribed tail decomposes at each step into extensions of $L^{n-4}$
-by $L^n$, and both have vanishing first cohomology for $n\ge 9$.
-
-\begin{proposition}[Finite compatible jets]\label{prop:jets}
-For the data of Proposition~\ref{prop:curve-data}, with the
-normalization of Proposition~\ref{prop:residue-data}, there exist
-finitely many holomorphic transition germs (each a composition of
-flows \eqref{eq:generator} of orders $1,\dots,7$) and holomorphic
-local rows $A_k,B_k$ for $0\le k\le 7$ such that:
-\begin{itemize}
-\item the transitions fix $C$ and preserve its conormal bundle $L$;
-\item the vector fields \eqref{eq:vector-fields} match through row
- index seven;
-\item $A\wedge B=t^2(1+O(t))$ on every chart.
-\end{itemize}
-On any surface neighborhood $Y$ realizing these transition germs, with
-$\cI=\OO_Y(-C)$ and $\cF=\cT_Y(-\log C)$, the rows give two sections
-of $\cF/\cI^9\cF$, both in $\cI\cF/\cI^9\cF$. Their determinant has
-leading term $t^4E_t\wedge\partial_z$, equivalently
-$t^5\partial_t\wedge\partial_z$, with coefficient $1$.
-\end{proposition}
-
-\begin{proof}
-Perform the first two orders and the induction through $k=7$. Only
-finitely many flows occur, each an actual holomorphic germ on an
-annular neighborhood of the zero section. Terms with row indices
-$0,\ldots,7$ have frame coefficients of degrees $1,\ldots,8$, and
-matching those terms is matching modulo $\cI^9\cF$. Finally
-\eqref{eq:determinant-form} and the degree-two determinant equation
-give
-\[
- v_1\wedge v_2=t^4(1+O(t))E_t\wedge\partial_z
- =t^5(1+O(t))\partial_t\wedge\partial_z.
-\]
-This statement uses only finite compatible jets.
-\end{proof}
-
-\section{Analytic realization and contraction}\label{sec:analytic}
-
-The finite jets of Section~\ref{sec:jets} live in overlapping charts.
-We now realize them on a single Hausdorff complex surface $Y$, lift
-them to holomorphic vector fields by cohomology vanishing, and contract
-$C$ to a singular point.
-
-\subsection{A Hausdorff surface neighborhood}\label{subsec:realization}
-
-\begin{lemma}\label{lem:realization}
-The overlap maps of Proposition~\ref{prop:jets} define, after shrinking
-their domains, a smooth Hausdorff complex surface $Y$ containing $C$.
-The conormal bundle is $L$, hence $C^2=-\deg L=-10$. The prescribed
-rows give two global sections
-$\bar v_1,\bar v_2\in H^0(Y,\cF/\cI^9\cF)$ with local logarithmic and
-ordinary determinants $t^4(1+O(t))$ and $t^5(1+O(t))$ in compatible
-coordinates.
-\end{lemma}
-
-\begin{proof}
-Let $N=L^{-1}$ be the split normal bundle. All changes to its
-disk-to-outside identifications are finite compositions of flows of
-holomorphic vector fields
-\[
- D=t^k\bigl(b(z)t\partial_t+h(z)\partial_z\bigr),\qquad k\ge 1,
-\]
-on punctured disk neighborhoods. On a compact annulus every $D$
-vanishes along the zero section; the local existence theorem for
-holomorphic differential equations, applied uniformly on the annulus,
-gives the time-one map and inverse after shrinking the fiber radius.
-There are finitely many such flows; successive shrinking produces
-their compositions on annular bands. Denote the resulting
-inside-to-outside maps by $\Phi_j$, $1\le j\le 6$.
-
-\emph{Quotient construction.} Choose mutually disjoint coordinate
-disks $D_j$ about the six $q_j$, and in each radii
-$0<a_j<a'_j<b'_j<b_j$ so that $\Phi_j$ and $\Phi_j^{-1}$ are defined
-on neighborhoods of the closed annulus $a_j\le |z_j|\le b_j$ in
-their split bundles. Retain a small open neighborhood $V_j$ of the
-zero section over $|z_j|<b'_j$ and a small open neighborhood $V_0$ of
-the zero section over $C\setminus\cup_j\{|z_j|\le a'_j\}$. Choose the
-transverse thicknesses small enough that $\Phi_j$ and $\Phi_j^{-1}$
-are defined throughout the retained parts of the closed annuli and
-\begin{equation}\label{eq:collar-bound}
- |z_j(\Phi_j(p))-z_j(p)|<\tfrac12(a'_j-a_j)
-\end{equation}
-there. This is possible because $\Phi_j$ restricts to the identity on
-the zero section. Make the retained annular images land over disjoint
-$D_j$.
-
-Set $A_j=\{p\in V_j:\,a_j<|z_j(p)|<b_j,\;\Phi_j(p)\in V_0\}$ and
-identify $p\in A_j$ with $\Phi_j(p)\in V_0$. These are biholomorphic
-identifications between open subsets. Each point of $V_0$ lies in at
-most one image because the retained annuli lie over disjoint $D_j$.
-Thus every equivalence class in $V_0\sqcup V_1\sqcup\cdots\sqcup V_6$
-has at most two elements: the equivalence relation consists exactly
-of the diagonal, these six graphs, and their inverses. No triple
-overlaps occur.
-
-\emph{Separation.} Each graph is closed relative to $V_j\times V_0$.
-Suppose $p_\nu\in A_j$, $p_\nu\to p\in V_j$, $\Phi_j(p_\nu)\to q\in V_0$.
-The base radii of the sources have limit in $[a_j,b'_j]$ with
-$b'_j<b_j$; the flows are defined across the closed annulus, so
-continuity gives $q=\Phi_j(p)$. If the source radius limit were $a_j$,
-inequality~\eqref{eq:collar-bound} would give $|z_j(q)|<a'_j$,
-contradicting $q\in V_0$. So $a_j<|z_j(p)|<b_j$, hence $p\in A_j$.
-Fiber boundary issues do not arise, since both limit points lie in
-the retained open sets.
-
-The equivalence relation is closed. The quotient map is open (gluing
-open sets by biholomorphisms), so the quotient is Hausdorff. Each
-$V_i$ maps injectively and openly into $Y$, giving $Y$ a smooth
-complex surface structure. The image of the zero section is $C$ with
-its original structure.
-
-\emph{Conormal bundle.} Each flow fixes $C$ and induces the identity
-on its conormal bundle. In the split identification the coordinate has
-$t'=t+O(t^2)$, so composing with the original split changes leaves
-$\cI/\cI^2\simeq L$. Hence $C^2=-\deg L=-10$.
-
-\emph{Jet sections.} In the logarithmic frame $(t\partial_t,\partial_z)$
-the rows $0,\ldots,7$ prescribe coefficients up to degree $8$ in $t$.
-The matching of Proposition~\ref{prop:jets} is matching modulo
-$\cI^9\cF$ on the actual overlaps. So the rows descend to global
-sections $\bar v_1,\bar v_2$ of $\cF/\cI^9\cF$. Their logarithmic
-determinant is $t^2(A\wedge B)=t^4(1+O(t))$. Replacing $t\partial_t$
-by $\partial_t$ multiplies by $t$, giving ordinary determinant
-$t^5(1+O(t))$.
-\end{proof}
-
-\subsection{Cohomology vanishing and lifting}\label{subsec:lifting}
-
-We now lift the jets to holomorphic vector fields.
-
-\begin{lemma}\label{lem:graded-vanishing}
-For every $n\ge 9$,
-\[
- H^1(C,\cF|_C\otimes L^n)=0.
-\]
-Consequently, with $\mathcal E=\cI^9\cF$ and every $r\ge 1$,
-\[
- H^1(Y,\mathcal E/\cI^r\mathcal E)=0.
-\]
-\end{lemma}
-
-\begin{proof}
-The logarithmic tangent sheaf is locally free of rank two and fits
-into the log restriction exact sequence
-\begin{equation}\label{eq:log-restriction}
- 0\to\OO_C\to\cF|_C\to\cT_C\to 0,
-\end{equation}
-with the first map sending $1$ to the normal Euler field $t\partial_t$
-and the second restricting a vector field to $C$. Since $K_C=L^4$,
-tensoring by $L^n$ gives
-\[
- 0\to L^n\to\cF|_C\otimes L^n\to L^{n-4}\to 0.
-\]
-Serre duality identifies the duals of the first cohomologies with
-\[
- H^0(C,L^{4-n})\quad\text{and}\quad H^0(C,L^{8-n}),
-\]
-both zero for $n\ge 9$ because $\deg L>0$. The cohomology sequence
-gives $H^1(C,\cF|_C\otimes L^n)=0$.
-
-Since $C$ is Cartier and $\cF$ is locally free,
-\[
- \cI^n\cF/\cI^{n+1}\cF\simeq\cF|_C\otimes L^n.
-\]
-The quotient $\mathcal E/\cI^r\mathcal E=\cI^9\cF/\cI^{9+r}\cF$ has
-a finite filtration with graded terms for $9\le n<9+r$, each with
-vanishing $H^1$ by the first part. Successive cohomology sequences
-give the second part.
-\end{proof}
-
-\begin{lemma}\label{lem:actual-lifting}
-After shrinking $Y$ about $C$, the sections $\bar v_1,\bar v_2$ lift
-to holomorphic sections $v_1,v_2$ of $\cF$ framing
-$\cT_{Y\setminus C}$.
-\end{lemma}
-
-\begin{proof}
-The intersection matrix of $C$ is the negative-definite $(-10)$. So
-Grauert's contraction~\cite[\S4.8(e), pp.~366--367]{Grauert1962}
-contracts $C$ to a point $x$ of a normal surface $X$ with
-\[
- \pi:Y\to X,\quad
- \pi^{-1}(x)=C\quad(\text{set}),
- \quad Y\setminus C\simeq X\setminus\{x\},
-\]
-and $\pi_*\OO_Y=\OO_X$. Choose a sufficiently small Stein representative
-$X$ about $x$ and replace $Y$ by its inverse image.
-
-\emph{$R^1\pi_*\mathcal E=0$.} Let $\mathfrak m_x$ be the ideal sheaf
-of $x$ in $X$ and $\mathcal J=\mathfrak m_x\OO_Y$. Both $\mathcal J$
-and $\cI$ vanish exactly on $C$; $\cI$ is radical. The analytic
-Nullstellensatz~\cite[Ch.~II, Thm.~4.22]{Demailly2012} and
-Noetherianity give $\cI^a\subseteq\mathcal J$ locally at each point
-of $C$; compactness of $C$ gives a single exponent $a$. After
-shrinking,
-\begin{equation}\label{eq:cofinal}
- \cI^a\subseteq\mathcal J\subseteq\cI,\qquad
- \cI^{ar}\subseteq\mathcal J^r\subseteq\cI^r
- \quad(r\ge 1).
-\end{equation}
-
-The proper analytic direct-image theorem makes
-$R^1\pi_*\mathcal E$ coherent. The Grauert theorem on formal
-functions~\cite[\S6, Haupts\"atze~I, IIa, p.~59]{Grauert1960}
-identifies the completed stalk with
-\begin{equation}\label{eq:formal-functions}
- \widehat{(R^1\pi_*\mathcal E)_x}
- \simeq\varprojlim_{r\geq 1}H^1(Y,\mathcal E/\mathcal J^r\mathcal E).
-\end{equation}
-More precisely, with the two ideal filtrations compared by
-\eqref{eq:cofinal}: the transition map from index $ar$ to index $r$ in
-$\varprojlim_r H^1(Y,\mathcal E/\mathcal J^r\mathcal E)$ factors
-through
-\[
- H^1(Y,\mathcal E/\cI^{ar}\mathcal E)=0
-\]
-by Lemma~\ref{lem:graded-vanishing}. The inverse system is pro-zero,
-so its inverse limit is zero. The finite $\OO_{X,x}$-module
-$(R^1\pi_*\mathcal E)_x$ has zero completion, hence is zero by
-Nakayama. Away from $x$ the map $\pi$ is an isomorphism, so
-$R^1\pi_*\mathcal E=0$ globally.
-
-\emph{Lifting.} The sheaf $\pi_*\mathcal E$ is coherent. Cartan's
-Theorem B on the Stein space $X$, together with the Leray spectral
-sequence for $\pi$, gives $H^1(Y,\mathcal E)=0$. The cohomology
-sequence of
-\[
- 0\to\mathcal E\to\cF\to\cF/\cI^9\cF\to 0
-\]
-lifts both $\bar v_i$ to holomorphic sections $v_i$ of $\cF$.
-
-\emph{Frame property.} In each divisor chart let
-$v_i^{(8)}$ be the polynomial representative of the prescribed jet.
-Then $v_i^{(8)}\in\cI\cF$ and $v_i-v_i^{(8)}\in\cI^9\cF$. Hence the
-logarithmic determinant of the pair differs from $t^4(1+O(t))$ by an
-element of $\cI^{10}\det\cF$; its leading coefficient is still $1$
-even at the common zeros of $r$. Ordinary determinant:
-\begin{equation}\label{eq:ordinary-determinant}
- v_1\wedge v_2=t^5u(t,z)\,\partial_t\wedge\partial_z,\qquad u(0,z)=1.
-\end{equation}
-The unit condition holds locally around each point of $C$; by
-compactness and properness, after further shrinking $X$ it holds on
-all of $Y$. So $v_1,v_2$ frame $\cT_{Y\setminus C}$.
-\end{proof}
-
-\subsection{The analytic singular point}\label{subsec:counterexample}
-
-\begin{proposition}\label{prop:analytic-counterexample}
-There is a normal complex surface germ $(X,x)$ with an isolated
-singularity and free analytic derivation sheaf of rank two:
-\[
- \cT_X=\mathcal H\mathrm{om}_{\OO_X}(\Omega^1_X,\OO_X)\simeq
- \OO_X^{\oplus 2}.
-\]
-Its resolution $Y\to X$ has exceptional set $C$, a smooth curve of
-genus $21$ and self-intersection $-10$.
-\end{proposition}
-
-\begin{proof}
-Take the contraction of Lemma~\ref{lem:actual-lifting}. Suppose $x$
-were smooth. A small neighborhood of $x$ would admit a
-nowhere-vanishing holomorphic two-form $\omega$; its pullback is
-holomorphic on $Y$ and non-vanishing off $C$. So
-\[
- \mathrm{div}(\pi^*\omega)=aC\quad(a\ge 0),\qquad
- K_Y\cdot C=aC^2=a\cdot(-10)\le 0.
-\]
-But adjunction gives
-\[
- K_Y\cdot C=2g(C)-2-C^2=40+10=50>0,
-\]
-a contradiction. So $x$ is singular, and it is the unique singular
-point after shrinking.
-
-Let $X^\circ=X\setminus\{x\}$ and $j:X^\circ\hookrightarrow X$. The
-frame on $X^\circ$ from Lemma~\ref{lem:actual-lifting} gives two
-holomorphic vector fields $w_1,w_2$ there. Normality in dimension two
-gives the Hartogs extension
-$\OO_X\simeq j_*\OO_{X^\circ}$~\cite[Ch.~II,
-Cor.~7.8]{Demailly2012}. This extends the $w_i$: embed $(X,x)$ in a
-smooth analytic germ with coordinates $x_1,\ldots,x_e$, extend the
-functions $w_i(x_\alpha)$ to functions $a_{i\alpha}$ on $X$, verify
-that
-$\sum_\alpha a_{i\alpha}\partial f/\partial x_\alpha=0$ for every
-local defining equation $f$ (by density and reducedness), and get
-sections of $\cT_X$. For any local derivation $w$ on $X$, the frame on
-$X^\circ$ writes $w=h_1w_1+h_2w_2$ there with holomorphic coefficients;
-both coefficients extend by Hartogs, giving the equality on $X$ by
-applying the difference to embedding coordinates. So
-$\OO_X^{\oplus 2}\to\cT_X$ is surjective. Injectivity: if
-$h_1w_1+h_2w_2=0$ on $X$, then both $h_i=0$ on $X^\circ$, hence on $X$.
-\end{proof}
-
-\begin{remark}\label{rem:invariants}
-The canonical discrepancy and a lower bound on $p_g$ follow from the
-construction. The two-form $\omega^\circ$ dual to the frame generates
-the canonical sheaf on $X^\circ$; reflexivizing across the normal
-point gives $\omega_X\simeq\OO_X$, so (choosing $K_X=0$) $X$ is
-Gorenstein. By \eqref{eq:ordinary-determinant},
-\[
- \pi^*\omega^\circ=t^{-5}u(t,z)^{-1}dt\wedge dz,
-\]
-which is meromorphic on $Y$ with divisor exactly $-5C$. So
-$K_Y=\pi^*K_X-5C$ and the discrepancy of $C$ is $-5$.
-
-For $p_g$, put $\OO_{nC}=\OO_Y/\cI^n$. From
-\eqref{eq:boundary-cohomology} and \eqref{eq:dimension-formula},
-$h^1(C,L)=12$. Then
-\[
- 0\to L\to\OO_{2C}\to\OO_C\to 0,
-\]
-with $H^2(C,L)=0$, gives
-$h^1(\OO_{2C})=h^1(C,L)+h^1(C,\OO_C)=12+21=33$. For every $n$ the
-sequence $0\to L^n\to\OO_{(n+1)C}\to\OO_{nC}\to 0$ shows
-$H^1(\OO_{(n+1)C})\to H^1(\OO_{nC})$ is surjective. Formal functions
-identifies their inverse limit with the completion of
-$(R^1\pi_*\OO_Y)_x$, which is a finite module supported at $x$;
-$p_g(X,x)\ge 33$. With $g=21$, $b=0$, we get $p_g-g-b\ge 12$.
-\end{remark}
-
-\section{Algebraization}\label{sec:algebra}
-
-The analytic germ $(X,x)$ from Section~\ref{sec:analytic} is what we
-want, but we need a finitely generated $\CC$-algebra. Approximating
-defining equations alone is insufficient: an approximation $\widetilde f$
-of $f$ need not have $P/(\widetilde f)$ isomorphic to $P/(f)$ if the
-relations between the $f_i$ are lost. The key is to approximate the
-equations and a finite relation matrix \emph{together}. This follows
-Hironaka's equation-and-relation determinacy~\cite[\S3,
-Thm.~3.3]{Hironaka1969}, used in Artin's algebraization of isolated
-singularities~\cite[Thms.~3.8--3.9]{Artin1969}. We prove the finite
-determinacy statement we need, including fixed estimates that avoid
-any complete-intersection hypothesis.
-
-\subsection{Finite determinacy of equations with relations}\label{subsec:finite-determinacy}
-
-Put
-\[
- P=\CC[[x_1,\ldots,x_e]],\quad
- \mathfrak n=(x_1,\ldots,x_e),\quad
- I=(f_1,\ldots,f_s),\quad R=P/I,
-\]
-with $f=(f_1,\ldots,f_s)\in\mathfrak n P^s$. Write
-$\mathfrak m_R=\mathfrak n R$, and use an overbar for reduction modulo
-$I$. Choose an $s\times t$ matrix $D$ whose columns generate all
-relations on $f$; thus
-\begin{equation}\label{eq:ideal-presentation}
- P^t\xrightarrow{D}P^s\xrightarrow{f}I\to 0
-\end{equation}
-is exact. Define
-\[
- \psi:R^s\to R^t,\quad v\mapsto v\overline D,\quad
- K=\ker\psi,\quad
- J=\sum_{j=1}^e R\overline{\partial_j f}.
-\]
-Differentiating $fD=0$ gives $J\subseteq K$. Right-exact reduction of
-\eqref{eq:ideal-presentation} gives a presentation of $I/I^2$, whose
-dual identifies
-\begin{equation}\label{eq:conormal-dual}
- K=\Hom_R(I/I^2,R),\qquad T^1=K/J.
-\end{equation}
-
-\begin{lemma}[Finite determinacy]\label{lem:finite-determinacy}
-Suppose $T^1$ in \eqref{eq:conormal-dual} has finite length. Then
-there exist integers $N,n_0\ge 0$ depending only on $f,D$ such that
-the following holds. Suppose a row $g\in P^s$ and an $s\times t$
-matrix $B$ satisfy
-\[
- gB=0,\quad g-f\in\mathfrak n^{n_0}P^s,\quad B-D\in\mathfrak n^N
- P^{s\times t}.
-\]
-Then there is a continuous $\CC$-algebra automorphism $\Phi$ of $P$
-and an invertible $U\in\mathrm{GL}_s(P)$ with
-\[
- f=\Phi(g)U,
-\]
-$\Phi(x_j)\equiv x_j\pmod{\mathfrak n^2}$ and $U\equiv 1\pmod{\mathfrak n}$.
-Consequently $P/(g)\simeq P/(f)$ as complete local $\CC$-algebras.
-\end{lemma}
-
-\begin{proof}
-We improve equations one order at a time. The exact relation $gB=0$
-makes the equation error, modulo fixed $I$, nearly lie in $K$; Artin--Rees
-lets us correct it into $K$; finite length of $K/J$ expresses the
-high-order corrected class by coordinate derivatives; a change of
-generators removes the remaining $I$-valued error. Corrections converge.
-
-\emph{Fixed bounds.} For a map $u:M\to M'$ of finite modules over a
-Noetherian local ring with maximal ideal $\mathfrak a$, the
-Artin--Rees lemma~\cite[Tag~00IN]{StacksApproximation} applied to
-$\mathrm{im}\,u\subset M'$ gives $a\ge 0$ with
-\[
- \mathrm{im}\,u\cap\mathfrak a^rM'\subset u(\mathfrak a^{r-a}M)
- \qquad(r\ge a).
-\]
-Choose nonnegative $\alpha,\beta,d$ such that
-\begin{align}
- \mathrm{im}\,\psi\cap\mathfrak m_R^rR^t&
- \subset\psi(\mathfrak m_R^{r-\alpha}R^s),
- \label{eq:ar-image}\\
- K\cap\mathfrak m_R^rR^s&\subset\mathfrak m_R^{r-\beta}K,
- \label{eq:ar-kernel}\\
- I^{\oplus s}\cap\mathfrak n^rP^s&
- \subset\{fC:C\in\mathfrak n^{r-d}P^{s\times s}\}
- \label{eq:ar-ideal}
-\end{align}
-for the relevant exponents. In \eqref{eq:ar-ideal}, Artin-Rees is
-applied to $P^{s\times s}\to P^s$, $C\mapsto fC$, with image $I^{\oplus s}$;
-the filtration on $I^{\oplus s}$ comes from $P^s$.
-
-Choose $\ell\ge 0$ with $\mathfrak m_R^\ell K\subset J$ (possible since
-$K/J=T^1$ has finite length). Put
-\begin{equation}\label{eq:constants}
- c_0=\beta+\ell,\quad c=c_0+d,\quad N\ge\alpha+1,\quad
- n_0\ge\max\{c+N,c+2,2c+1\}.
-\end{equation}
-
-\emph{One improvement step.} Suppose $n\ge n_0$ and
-$h=g-f\in\mathfrak n^nP^s$, $B-D\in\mathfrak n^NP^{s\times t}$, $gB=0$.
-
-The exact relation, reduced modulo fixed $I$, gives
-\begin{equation}\label{eq:high-order-relation}
- \bar h\bar D=\bar g\bar D=-\bar g(\bar B-\bar D)
- =-\bar h(\bar B-\bar D)\in\mathfrak m_R^{n+N}R^t.
-\end{equation}
-The factor $\bar h$ on the right gives order $n+N$.
-By \eqref{eq:ar-image} there is
-\[
- \bar\epsilon\in\mathfrak m_R^{n+N-\alpha}R^s
- \subset\mathfrak m_R^{n+1}R^s\quad\text{with}\quad
- \psi(\bar\epsilon)=\psi(\bar h).
-\]
-So $\kappa=\bar h-\bar\epsilon\in K\cap\mathfrak m_R^nR^s\subset
-\mathfrak m_R^{n-\beta}K\subset\mathfrak m_R^{n-c_0}J$.
-Lift coefficients of $\kappa$ in generators of $J$ to
-$a_j\in\mathfrak n^{n-c_0}$; lift $\bar\epsilon$ to
-$\epsilon\in\mathfrak n^{n+1}P^s$. Then
-\[
- w=h-\sum_{j=1}^e a_j\partial_jf-\epsilon
- \in I^{\oplus s}\cap\mathfrak n^{n-c_0}P^s.
-\]
-Using \eqref{eq:ar-ideal}, $w=fC$ with $C\in\mathfrak n^{n-c}P^{s\times s}$.
-Thus
-\begin{equation}\label{eq:linear-correction}
- h=\sum_{j=1}^e a_j\partial_jf+fC+\epsilon,\quad
- a_j,C\in\mathfrak n^{n-c},\quad\epsilon\in\mathfrak n^{n+1}P^s.
-\end{equation}
-
-Put $q=n-c$ and define
-\begin{equation}\label{eq:one-step}
- \phi(x_j)=x_j-a_j,\quad M=(1+C)^{-1},\quad
- g_{\mathrm{new}}=\phi(g)M,\quad B_{\mathrm{new}}=(1+C)\phi(B).
-\end{equation}
-The bounds give $q\ge\max\{N,2\}$, $2q\ge n+1$, $n+q-1\ge n+1$.
-In particular $\phi$ is a formal automorphism with identity linear
-part and $M$ is invertible. Taylor expansion gives
-\[
- \phi(f)-f+\sum a_j\partial_jf\in\mathfrak n^{2q}P^s,\qquad
- \phi(h)-h\in\mathfrak n^{n+q-1}P^s.
-\]
-Combining with \eqref{eq:linear-correction}:
-\[
- \phi(g)\equiv f(1+C)\pmod{\mathfrak n^{n+1}P^s},\quad
- g_{\mathrm{new}}-f\in\mathfrak n^{n+1}P^s.
-\]
-And
-$g_{\mathrm{new}}B_{\mathrm{new}}=\phi(g)\phi(B)=\phi(gB)=0$.
-For $B$: substitution $\phi$ changes any power series by an element
-of $\mathfrak n^q$, and $C\in\mathfrak n^q$, so
-\[
- B_{\mathrm{new}}-D=C\phi(B)+(\phi(B)-B)+(B-D)\in\mathfrak n^NP^{s\times t}.
-\]
-
-\emph{Iteration.} Starting from $(g_{n_0},B_{n_0})=(g,B)$, repeat for
-$n=n_0,n_0+1,\dots$. Set
-$\Phi_{n_0}=\id$, $U_{n_0}=1$,
-$\Phi_{n+1}=\phi_n\circ\Phi_n$,
-$U_{n+1}=\phi_n(U_n)M_n$, so that $g_n=\Phi_n(g)U_n$.
-Since $(\phi_n-\id)(P)\subset\mathfrak n^{n-c}$ and
-$M_n-1\in\mathfrak n^{n-c}P^{s\times s}$, consecutive images
-$\Phi_n(x_j)$ and matrices $U_n$ differ in orders tending to $\infty$;
-they converge in $P$ to $\Phi(x_j)$ and $U$. Then $\Phi$ has identity
-linear part (so is an automorphism), $U\equiv 1\pmod{\mathfrak n}$ (so
-is invertible), and passing to the limit in $g_n=\Phi_n(g)U_n$ gives
-$f=\Phi(g)U$.
-\end{proof}
-
-\begin{remark}\label{rem:finite-determinacy-note}
-The original $D$ generates all relations, so \eqref{eq:conormal-dual}
-holds. The approximating $B$ need only satisfy the exact zero product
-and be close to $D$; no relation generation is required. So
-Lemma~\ref{lem:finite-determinacy} concerns solutions of $gB=0$, not
-arbitrary truncations of the defining equations.
-\end{remark}
-
-\subsection{The algebraization proposition}\label{subsec:algebraization}
-
-\begin{proposition}[Algebraization]\label{prop:algebraization}
-Let $(X,x)$ be a normal complex analytic germ of dimension $d\ge 1$,
-smooth away from $x$, with holomorphic tangent module free of rank $d$.
-There exist a finitely generated normal integral $\CC$-algebra $A$
-and a maximal ideal $\mathfrak p\subset A$ such that
-\[
- \dim A=d,\quad A/\mathfrak p=\CC,\quad
- \widehat{A_{\mathfrak p}}\simeq\widehat{\OO_{X,x}},\quad
- \Der_\CC(A)\simeq A^d.
-\]
-The variety $\Spec A$ can be chosen smooth away from $\mathfrak p$.
-If $x$ is singular, $A_{\mathfrak p}$ is nonregular.
-\end{proposition}
-
-\begin{proof}
-\emph{Setup.} Write $S=\OO_{X,x}=\CC\{x_1,\ldots,x_e\}/I_{\mathrm{an}}$.
-Choose a finite row $f$ generating $I_{\mathrm{an}}$ and a finite
-matrix $D$ generating all its relations over the convergent ring.
-Completion is exact on finite modules, so they give
-\eqref{eq:ideal-presentation} over $P=\CC[[x_1,\ldots,x_e]]$, with
-$R=\widehat S=P/(f)$. Complex analytic local rings are
-excellent~\cite[Prop.~2.1(1)]{Cutkosky2017}, so the normal local
-ring $S$ completes to a normal local domain $R$; also
-\begin{equation}\label{eq:completion-invariants}
- \dim R=\dim S=d,\quad
- R\text{ regular }\Leftrightarrow S\text{ regular}.
-\end{equation}
-
-\emph{$T^1$ has finite length.} On the smooth locus, the
-tangent-to-normal map is surjective (a smooth subspace is locally
-regular); its cokernel is a coherent module supported at the isolated
-$x$, hence finite length. Completion preserves this finite-length
-support, yielding the analogue of \eqref{eq:conormal-dual} for $f,D$
-with finite-length $T^1$. So Lemma~\ref{lem:finite-determinacy}
-applies.
-
-\emph{Tangent modules.} The holomorphic differentials of $S$ have a
-finite Jacobian presentation; its dual is the tangent module, which is
-free of rank $d$ by hypothesis. Flat completion of the kernel gives
-\begin{equation}\label{eq:continuous-tangent}
- \Der_\CC^{\mathrm{cont}}(R,R)
- =\ker\bigl(R^e\to R^s,\,(v_j)\mapsto\sum_j v_j\overline{\partial_jf}
- \bigr)\simeq R^d.
-\end{equation}
-A continuous derivation of a power-series quotient is determined by
-values on $x_j$, and the displayed relations are exactly the descent
-conditions.
-
-\emph{Approximation.} Let
-$H=(\CC[x_1,\ldots,x_e]_{(x_1,\ldots,x_e)})^h$ be the henselization;
-its completion is $P$. Apply Artin's approximation theorem over $H$
-to the system of $t$ equations
-$\sum_i G_iB_{i\nu}=0$ ($1\le\nu\le t$) in the $s+st$ entries of
-$G\in H^s$, $B\in H^{s\times t}$, with formal solution $(f,D)$
-\begin{equation}\label{eq:artin-system}
- \sum_{i=1}^s G_iB_{i\nu}=0\quad(1\le\nu\le t).
-\end{equation}
-Get $g,B$ with $gB=0$ exactly and $g\equiv f$, $B\equiv D$ to any
-order~\cite[Thm.~1.10]{Artin1969}, \cite[Tags~07QY, 07QZ]{StacksApproximation}.
-Take that order $\ge\max\{n_0,N\}$ with $n_0,N$ from
-Lemma~\ref{lem:finite-determinacy}; the lemma then gives
-\begin{equation}\label{eq:formal-equivalence}
- P/(g)\simeq R.
-\end{equation}
-
-\emph{Spreading to a finite-type neighborhood.} Henzelization is the
-filtered colimit of pointed \'etale neighborhoods. So the finitely
-many entries of $g,B$ are represented on a single such neighborhood,
-and after passing to a later one the identities
-\eqref{eq:artin-system} hold there as well. Expressing this stage
-over the local polynomial ring, finite presentation and clearing
-denominators spread it to an affine \'etale neighborhood of the origin
-in $\mathbb A^e_\CC$.
-
-The result is an affine \'etale map
-$\mathrm{Spec}\,E\to\mathbb A^e_\CC$, a marked point $\mathfrak q$
-over the origin with residue field $\CC$, and a row $g$ of $E$ with
-$g(\mathfrak q)=0$. The completed ambient local ring is $P$. Put
-$A'=E/(g)$, $\mathfrak p'=\mathfrak q/(g)$, $T=A'_{\mathfrak p'}$.
-Then $A'$ is finitely generated over $\CC$, and exactness of
-completion plus \eqref{eq:formal-equivalence} gives
-\begin{equation}\label{eq:completed-local-ring}
- \widehat T=P/(g)\simeq R.
-\end{equation}
-
-\emph{Descent of tangent freeness.} Because $E$ is \'etale over
-$\mathbb A^e_\CC$, its differentials have basis $dx_1,\ldots,dx_e$
-near $\mathfrak q$. Completing the finite Jacobian presentation of
-$\Omega^1_{T/\CC}$ produces the continuous presentation of $P/(g)$.
-For a finitely presented module, flat base change commutes with its
-dual (express the dual as a kernel of finite-free maps). Chasing
-through \eqref{eq:completed-local-ring} and
-\eqref{eq:continuous-tangent}:
-\begin{equation}\label{eq:tangent-descent}
- \Der_\CC(T)\otimes_T\widehat T
- \simeq\Der_\CC^{\mathrm{cont}}(\widehat T,\widehat T)
- \simeq\widehat T^d.
-\end{equation}
-Let $M=\Der_\CC(T)$. From \eqref{eq:tangent-descent}, the residue
-vector space of $M$ is $d$-dimensional. Lift a basis to $d$ elements
-of $M$: Nakayama gives a surjection $T^d\to M$, which after
-completion is an isomorphism of free rank-$d$ modules and hence
-iso; faithful flatness of completion makes the kernel zero, so
-$M\simeq T^d$.
-
-\emph{Local ring properties.} $T\to\widehat T$ is injective and
-$\widehat T$ is a domain; so $T$ is a domain. For normality: if
-$a/b\in\mathrm{Frac}(T)$ is integral over $T$, it is integral over
-the normal domain $\widehat T$, hence lies in $\widehat T$. Then
-$a\in b\widehat T\cap T=bT$ (last equality by faithful flatness),
-so $a/b\in T$. Completion also preserves Krull dimension and
-$1+\dim_{\CC(\mathfrak p)}\mathfrak p/\mathfrak p^2$; by
-\eqref{eq:completion-invariants} and \eqref{eq:completed-local-ring},
-$T$ has dimension $d$ and is nonregular whenever $S$ is.
-
-\emph{Isolated singularity.} Let $\mathfrak j_{\mathrm{an}}$ be the
-$d$th Fitting ideal of the holomorphic differential module of $S$.
-The analytic Jacobian criterion identifies its zero set with the
-nonsmooth locus, so some power of $\mathfrak m_S$ lies in
-$\mathfrak j_{\mathrm{an}}$. Fitting ideals commute with base change.
-The Jacobian presentations above and \eqref{eq:completed-local-ring}
-give, by downward contraction from faithful completion,
-\begin{equation}\label{eq:isolated-descent}
- \mathfrak m_T^b\subset\mathrm{Fitt}_d(\Omega^1_{T/\CC})
- \quad\text{for some }b\ge 0.
-\end{equation}
-On the reduced pure-$d$-dimensional neighborhood constructed next,
-the algebraic Jacobian criterion identifies this containment as
-nonsmoothness supported only at $\mathfrak p$.
-
-\emph{Affine neighborhood.} $\Der_\CC(A')$ localizes at $\mathfrak p'$
-to $\Der_\CC(T)$ because $\Omega^1_{A'/\CC}$ is finitely presented.
-The chosen basis spreads after inverting an element outside
-$\mathfrak p'$: it defines a map free-to-derivation whose kernel and
-cokernel vanish at $\mathfrak p'$, so a further principal shrinking
-makes the map an isomorphism.
-
-Since $T$ is a domain, exactly one irreducible component of
-$\Spec A'$ passes through $\mathfrak p'$, and the support of the
-nilradical avoids $\mathfrak p'$. Remove the other components and
-that support. The remaining neighborhood is reduced and irreducible.
-Its normal locus is open (finite type over $\CC$ is excellent) and
-contains $\mathfrak p'$ by normality of $T$; restrict there. The
-irreducible neighborhood has dimension $d$. Its nonsmooth locus is
-closed; by \eqref{eq:isolated-descent} it has no positive-dimensional
-component through $\mathfrak p'$; remove the components that do not
-contain $\mathfrak p'$.
-
-All the restrictions above are open neighborhoods of $\mathfrak p'$.
-Choose a principal affine neighborhood contained in their
-intersection, write it $\Spec A$ with marked maximal
-$\mathfrak p$. Then $A$ is finitely generated, normal, integral,
-dimension $d$; $A_{\mathfrak p}=T$; $\Der_\CC(A)\simeq A^d$; residue
-field and completed local ring are unchanged; the variety is smooth
-away from $\mathfrak p$. This proves all the assertions.
-\end{proof}
-
-\begin{proof}[Proof of Theorem~\ref{thm:main}]
-Proposition~\ref{prop:analytic-counterexample} gives a singular
-normal complex analytic surface germ, smooth away from its marked
-point, with free holomorphic tangent module of rank two. Apply
-Proposition~\ref{prop:algebraization}; its coordinate ring $A$ is
-finitely generated, normal, integral over $\CC$, dimension $2$, with
-$\Der_\CC(A)\simeq A^2$ and $A_{\mathfrak p}$ nonregular. So the
-derivation module is free while $A$ is not regular. This is the
-required counterexample over $\CC$.
-\end{proof}
-
-\section{Geometric consequences}\label{sec:consequences}
-
-The counterexample carries secondary structure worth recording.
-
-\subsection*{Logarithmic Kodaira dimension}
-
-For a smooth quasiprojective surface $U$ with smooth projective
-compactification $\overline U$ and simple normal-crossing boundary
-$D=\overline U\setminus U$, the logarithmic Kodaira dimension is
-\[
- \overline\kappa(U)=\kappa(K_{\overline U}+D).
-\]
-This value does not depend on the compactification.
-
-\begin{corollary}\label{cor:log-kodaira}
-The smooth locus $U=X_{\mathrm{reg}}$ of the affine surface
-$X=\Spec A$ in Theorem~\ref{thm:main} has $\overline\kappa(U)=2$.
-\end{corollary}
-
-\begin{proof}
-The free tangent sheaf on $X$ restricts to a trivial tangent bundle on
-$U$; the top exterior power of that bundle gives a nowhere-vanishing
-bivector on $U$, whose dual is a nowhere-vanishing two-form on $U$.
-If $\overline\kappa(U)\le 1$, the theorem of Biswas, Gurjar, and
-Kolte~\cite[Theorem~0.1(1)]{BiswasGurjarKolte2014} makes the normal
-affine surface $X$ smooth, contrary to Theorem~\ref{thm:main}. The
-only remaining possible value for a surface is two.
-\end{proof}
-
-\subsection*{Failure of derivation blowups to resolve}
-
-For a finite $A$-module $M$ of generic rank $r$, viewed as a coherent
-sheaf on $X$, the blowup at $M$ is universal among proper birational
-morphisms from integral varieties to $X$ for which the pullback of $M$
-modulo torsion is locally free of rank $r$. Every such morphism factors
-uniquely through the blowup; see~\cite[\S1.6, Thm.~1.17]{BarajasChavezRomano2026}.
-
-\begin{corollary}\label{cor:derivation-blowups}
-Starting with the singular surface $X=\Spec A$ of
-Theorem~\ref{thm:main}, every iteration of blowing up at the
-derivation module is isomorphic to $X$. The same holds if each
-blowup is followed by normalization. Neither procedure resolves the
-singular point.
-\end{corollary}
-
-\begin{proof}
-The derivation module is free. Its blowup is the identity
-morphism. The same holds after composition with normalization, since
-$X$ is already normal.
-\end{proof}
-
-\section*{Conclusion}
-
-The surface $X$ of Theorem~\ref{thm:main} is normal, singular at
-exactly one point, and its module of algebraic vector fields is free
-of rank two. This is a counterexample to the Lipman--Zariski
-conjecture in characteristic zero. The method combines six features
-that seem to be forced by the problem:
-\begin{enumerate}
-\item Algebraic construction of a curve with fourth root of $K_C$ and
- prescribed residue data (Section~\ref{sec:curve}).
-\item A trace pairing that transports quadratic structures from an
- auxiliary elliptic curve to the plane curve (Section~\ref{sec:trace}).
-\item Successive gluing of vector-field jets in which the leading
- determinant coefficient is arranged globally while the vertical
- obstruction is measured by Serre duality and cancelled by residue
- identities (Section~\ref{sec:jets}).
-\item Realization of finite jets on a Hausdorff neighborhood and
- lifting to holomorphic fields by cohomology vanishing
- (Section~\ref{sec:analytic}).
-\item Contraction to a normal surface point at which the exceptional
- curve has self-intersection $-10$ (Lemma~\ref{lem:actual-lifting}).
-\item Algebraization via simultaneous approximation of equations and a
- finite relation matrix, faithful descent of the tangent module,
- and affine shrinking (Section~\ref{sec:algebra}).
-\end{enumerate}
-
-The construction also shows that:
-\begin{itemize}
-\item The frame cannot be chosen commuting (J\"order).
-\item Iterated blowup at the derivation module does not resolve the
- singular point.
-\item The smooth locus has logarithmic Kodaira dimension two.
-\item The canonical discrepancy of the exceptional curve is $-5$;
- the geometric genus at the singular point satisfies $p_g\ge 33$
- (the bound $p_g-g-b\ge 12$ for a normal singularity resolved on one
- side by a curve of genus $g=21$ and $b=0$).
-\end{itemize}
-
-\section*{Acknowledgements}
-
-The author thanks the open literature for the many positive results
-that focused attention on the missing case of surfaces in
-characteristic zero.
-
-\begin{thebibliography}{99}\small
-
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-
-\bibitem{BarajasChavezRomano2026}
-A.~Barajas and P.~Ch\'avez and M.~Romano,
-\emph{Blowups at modules and their resolutions},
-in preparation, 2026.
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-\bibitem{BergnerGraf2020}
-H.~Bergner and P.~Graf,
-\emph{The Lipman--Zariski conjecture in genus one higher},
-Forum Math. Sigma \textbf{8} (2020), e21, 16~pp.
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-
-\bibitem{BiswasGurjarKolte2014}
-I.~Biswas and R.~V.~Gurjar and A.~Kolte,
-\emph{Normal affine surfaces with trivial tangent bundle},
-Math. Z. \textbf{278} (2014), 339--345.
-
-\bibitem{Cartan1952}
-H.~Cartan,
-\emph{S\'eminaire Cartan, Fonctions analytiques de plusieurs variables},
-Secr\'etariat math\'ematique, Paris, 1952.
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-\bibitem{Cutkosky2017}
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-Grad. Stud. Math.\ \textbf{63}, Amer. Math. Soc., 2004.
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-\bibitem{Demailly2012}
-J.-P.~Demailly,
-\emph{Complex Analytic and Differential Geometry},
-open access book, \url{https://www-fourier.ujf-grenoble.fr/~demailly/books.html}, 2012.
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-\bibitem{Druel2014}
-S.~Druel,
-\emph{The Zariski--Lipman conjecture for log canonical spaces},
-Bull. LMS \textbf{46} (2014), 827--835.
-\href{https://arxiv.org/abs/1304.6000}{arXiv:1304.6000}.
-
-\bibitem{FlennerZaidenberg2001}
-H.~Flenner and M.~Zaidenberg,
-\emph{Normal affine surfaces with trivial tangent bundle},
-J. Reine Angew. Math. \textbf{535} (2001), 1--61.
-
-\bibitem{GKKP2011}
-D.~Greb and S.~Kebekus and S.~J.~Kov\'acs and T.~Peternell,
-\emph{Differential forms on log canonical spaces},
-Publ. Math. IH\'ES \textbf{114} (2011), 87--169.
-\href{https://arxiv.org/abs/1003.5340}{arXiv:1003.5340}.
-
-\bibitem{GrafKovacs2014}
-P.~Graf and S.~J.~Kov\'acs,
-\emph{An optimal extension theorem for $1$-forms and the Lipman--Zariski conjecture},
-Doc. Math. \textbf{19} (2014), 815--830.
-\href{https://doi.org/10.4171/DM/465}{doi:10.4171/DM/465}.
-
-\bibitem{GrafLowGenus2021}
-P.~Graf,
-\emph{The Lipman--Zariski conjecture in low genus},
-Int. Math. Res. Not. \textbf{2021}, no.~1, 426--441.
-\href{https://arxiv.org/abs/1801.05753v2}{arXiv:1801.05753v2}.
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-\bibitem{GrafPositive2022}
-P.~Graf,
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-characteristic},
-J. Algebra \textbf{601} (2022), 115--128.
-\href{https://arxiv.org/abs/2201.06646v2}{arXiv:2201.06646v2}.
-
-\bibitem{Grauert1960}
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-\emph{Ein Theorem der analytischen Garbentheorie und die
-Modulr\"aume komplexer Strukturen},
-Publ. Math. IH\'ES \textbf{5} (1960), 5--64.
-\href{https://doi.org/10.1007/BF02684746}{doi:10.1007/BF02684746}.
-
-\bibitem{Grauert1962}
-H.~Grauert,
-\emph{\"Uber Modifikationen und exzeptionelle analytische Mengen},
-Math. Ann. \textbf{146} (1962), 331--368.
-\href{https://doi.org/10.1007/BF01441136}{doi:10.1007/BF01441136}.
-
-\bibitem{GrauertRemmert1984}
-H.~Grauert and R.~Remmert,
-\emph{Coherent Analytic Sheaves},
-Grundlehren Math. Wiss.\ \textbf{265}, Springer, 1984.
-
-\bibitem{Grothendieck1957}
-A.~Grothendieck,
-\emph{Sur la classification des fibr\'es holomorphes sur la
-sph\`ere de Riemann},
-Amer. J. Math. \textbf{79} (1957), 121--138.
-\href{https://doi.org/10.2307/2372388}{doi:10.2307/2372388}.
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-\bibitem{Hironaka1969}
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-\emph{Formal line bundles along exceptional loci},
-in \emph{Algebraic Geometry} (Bombay Colloquium, 1968),
-Tata Inst. Fund. Res. Stud. Math., vol.~4, Oxford Univ. Press, 1969,
-201--218.
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-\bibitem{Hochster1977}
-M.~Hochster,
-\emph{The Zariski--Lipman conjecture in the graded case},
-J. Algebra \textbf{47} (1977), no.~2, 411--424.
-\href{https://doi.org/10.1016/0021-8693(77)90232-0}{doi:10.1016/0021-8693(77)90232-0}.
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-\bibitem{Jorder2014}
-C.~J\"order,
-\emph{A weak version of the Lipman--Zariski conjecture},
-Math. Z. \textbf{278} (2014), 893--899.
-\href{https://arxiv.org/abs/1311.5141v1}{arXiv:1311.5141v1}.
-
-\bibitem{Kallstrom2011}
-R.~K\"allstr\"om,
-\emph{The Zariski--Lipman conjecture for complete intersections},
-J. Algebra \textbf{337} (2011), no.~1, 169--180.
-\href{https://arxiv.org/abs/1003.4241v6}{arXiv:1003.4241v6}, 2018.
-
-\bibitem{Lipman1965}
-J.~Lipman,
-\emph{Free derivation modules on algebraic varieties},
-Amer. J. Math. \textbf{87} (1965), no.~4, 874--898.
-\href{https://doi.org/10.2307/2373252}{doi:10.2307/2373252}.
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-\emph{Differentielle Eigenschaften der Lokalisierungen analytischer
-Algebren},
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-J.-P.~Serre,
-\emph{G\'eom\'etrie alg\'ebrique et g\'eom\'etrie analytique},
-Ann. Inst. Fourier \textbf{6} (1956), 1--42.
-
-\bibitem{StacksApproximation}
-The Stacks Project Authors,
-\emph{The Stacks Project},
-\href{https://stacks.math.columbia.edu/tag/07QY}{Tag~07QY}
-(approximation over henselian local G-rings),
-\href{https://stacks.math.columbia.edu/tag/07QZ}{Tag~07QZ}
-(approximation in pointed \'etale neighborhoods).
-
-\bibitem{StacksCompletion}
-The Stacks Project Authors,
-\emph{The Stacks Project},
-\href{https://stacks.math.columbia.edu/tag/00IN}{Tag~00IN}
-(Artin--Rees),
-\href{https://stacks.math.columbia.edu/tag/00MA}{Tag~00MA}
-(exactness of completion on finite modules),
-\href{https://stacks.math.columbia.edu/tag/00MC}{Tag~00MC}
-(faithful flatness of completion),
-\href{https://stacks.math.columbia.edu/tag/07NV}{Tag~07NV}
-(dimension preserved by completion),
-\href{https://stacks.math.columbia.edu/tag/07NY}{Tag~07NY}
-(regularity preserved by completion),
-\href{https://stacks.math.columbia.edu/tag/0C23}{Tag~0C23}
-(normality preserved by completion).
-
-\bibitem{VanStratenSteenbrink1985}
-D.~van Straten and J.~Steenbrink,
-\emph{Extendability of holomorphic differential forms near isolated
-hypersurface singularities},
-Abh. Math. Sem. Univ. Hamburg \textbf{55} (1985), 97--110.
-
-\end{thebibliography}
-
-\end{document}
+
+ X^3 - b X^2 + b lambda X - lambda = 0.
+
+Vieta gives sum z_i = b, sum 1/z_i = b, and 2 sum_i d(z_i) = (h,h).
+Taking each d(z_i) twice produces the required vectors.
+
+The three directions are distinct. As lambda -> 0, two roots go to
+zero and the third goes to b. The roots are distinct for general
+lambda: the cubic discriminant has nonzero leading term
+-4 b^3 lambda. Near zero, the projective ratio of the entries of d(z)
+is z(R z - h/2)/(R - h z/2), whose derivative at zero is
+-h/(2R) != 0. The two small roots give distinct nonisotropic
+directions. At z = b, the entries are 3h/2 and (R^2 - h^2)/(2h),
+both nonzero, so the third direction differs from their common
+limit. Apply simultaneously to all six vectors the isometry
+(x,y) -> (nu x, nu^{-1} y). On nonisotropic directions the ratio is
+multiplied by nu^2, so only finitely many values of nu != 0 are
+excluded by the prescribed finite set; all required equations are
+preserved. []
+
+Apply the lemma to Q, excluding the branch values of the Phi-map. Each
+of the three directions is [ev_q] at two distinct points of Gamma.
+Label the six points q_1, ..., q_6 so [w_j] = [ev_{q_j}], and put
+
+ Z = q_1 + ... + q_6 in |3 Phi| = |M|.
+
+Choose a section beta of M with divisor Z. Nonisotropy implies
+F|_Gamma is unramified at every q_j. Their F-values are not all
+equal, since a degree-two fiber cannot contain six distinct points.
+
+The curve C is constructed with fourth-order contact with Gamma at
+every q_j. Let gamma denote the section of O_Sigma(Gamma) defining
+Gamma. Choose a section
+
+ kappa in H^0(Sigma, 2 Gamma + 4 F), kappa|_Gamma = beta^4,
+
+using the fixed identification above. Such sections exist because
+restriction to Gamma has kernel O_Sigma(Gamma + 4 F) = O_Sigma(6,2),
+with H^1 = 0. The affine space of choices is
+kappa_0 + gamma H^0(Sigma, Gamma + 4F). A general choice defines a
+smooth curve C: the linear system is base-point-free off Gamma, so
+Bertini gives smoothness there; on Gamma minus Z every choice is
+nonzero; and at each q_j, global generation of O_Sigma(6,2) makes the
+normal derivative vary nontrivially. Avoiding six proper affine
+hyperplanes makes those derivatives nonzero. Independence of the six
+evaluation functionals is unnecessary.
+
+Explicitly, near q_j choose a local equation y = 0 for Gamma, a
+coordinate z on Gamma vanishing at q_j, and frames in which kappa
+has coefficient k(y,z) and beta has coefficient b(z). The frames
+may be chosen compatibly so that k(0,z) = b(z)^4, b'(0) != 0, and
+k_y(0,0) != 0. The implicit function theorem writes C locally as
+y = phi(z), where
+
+ phi(z) = - b'(0)^4 / k_y(0,0) * z^4 + O(z^5).
+
+Thus the contact order is exactly four, div_C(gamma|_C) = 4 Z, and
+the six marked points are read both on Gamma and on C.
+
+The bidegree of C is (8,4). The defining-divisor sequence, with
+H^1(Sigma, O_Sigma(-8,-4)) = 0, gives H^0(C, O_C) = C, so C is
+connected and irreducible. Put
+
+ H = F|_C, L = H + Z, theta = 2L,
+
+and let xi be the canonical section of O_C(Z). The divisor equality
+specifies the isomorphism 4 O_C(Z) -> O_C(Gamma) by xi^4 -> gamma|_C.
+Together with adjunction this gives
+
+ 4L = 4H + 4Z = (Gamma + 4F)|_C = K_C.
+
+The last identification is normalized by the requirement that a local
+section s of Gamma + 4F on Sigma maps to
+
+ s|_C -> Res_C( s gamma Omega / kappa ),
+
+with the normal differential first. This fixes the scalar in the
+canonical-root isomorphism for subsequent residue computations.
+
+The next lemma computes the section spaces that transfer from Sigma to
+C and to Gamma. It uses the comparison theorem of Serre (Ann. Inst.
+Fourier 6, 1956, 1-42): algebraic and analytic section spaces of
+coherent sheaves on projective complex curves agree.
+
+./serre-1956-gaga.pdf
+
+Lemma (adjoint jets). For 0 <= j <= 3, put A_j = Gamma + (4-j)F =
+(6-j, 2) on Sigma and N_j = A_j|_Gamma = (8-j)F + Delta on Gamma.
+Restriction identifies H^0(Sigma, A_j) with H^0(C, K_C - jH) and
+maps it onto H^0(Gamma, N_j). For 1 <= ell <= 4, evaluation on the
+length-ell subschemes at q_1, ..., q_6 has the same rank on C as on
+Gamma. The ranks needed below are:
+
+ j = 0, ell = 1: h^0(Gamma, N_0) = 16; N_0 - Z = 5F;
+ h^0(Gamma, 5F) = 10; rank = 6.
+ j = 1, ell = 2: h^0(Gamma, N_1) = 14; N_1 - 2Z = Phi;
+ h^0(Gamma, Phi) = 2; rank = 12.
+ j = 2, ell = 2: h^0(Gamma, N_2) = 12; N_2 - 2Z = Delta;
+ h^0(Gamma, Delta) = 0; rank = 12.
+ j = 3, ell = 2: h^0(Gamma, N_3) = 10; N_3 - 2Z = -F + Delta;
+ h^0(Gamma, -F + Delta) = 0; rank = 10.
+
+The identities use 2Z = 6F, valid because Delta = 2D_0 has order 2.
+
+Proof. The restriction kernels on Sigma are A_j - C = (-2-j, -2)
+and A_j - Gamma = (4-j, 0). Product cohomology on PP^1 x PP^1 gives
+H^0 = H^1 = 0 for the first kernel and H^1 = 0 for the second, so
+restriction H^0(Sigma, A_j) -> H^0(C, K_C - jH) is an isomorphism and
+H^0(Sigma, A_j) -> H^0(Gamma, N_j) is onto. By the contact expansion,
+the ideal of the length-ell subscheme at q_j in Sigma equals
+(y - phi(z), z^ell) = (y, z^ell) for ell <= 4, and the same ideal
+defines the length-ell subscheme on C. So the two subschemes agree,
+and the ranks of evaluation agree.
+
+On the elliptic curve deg F = 2 and Z = 3F + Delta. Positive-degree
+bundles have as many sections as their degree; the nonzero 2-torsion
+Delta has none; negative-degree bundles have none. Subtracting the
+residual section dimension from h^0(Gamma, N_j) gives the stated
+evaluation ranks. []
+
+Proposition (curve data). There exist a smooth connected projective
+curve C, a reduced divisor Z = q_1 + ... + q_6 on C, line bundles H,
+L, theta on C, and a basis u = (u_1, u_2) of H^0(C, H) with:
+
+(1) g(C) = 21, deg H = 4, deg L = 10, L = H + Z, theta = 2L, and
+K_C = 4L under the fixed normalized isomorphism.
+
+(2) The pencil u defines a finite degree-four morphism f: C -> PP^1
+unramified at every q_j, with f(q_j) not all equal. The zero divisor
+of u_1 can be chosen to be a reduced fiber disjoint from Z.
+
+(3) Section dimensions and multiplication:
+ h^0(C, jH) = j+1 (0 <= j <= 3),
+ h^0(C, L) = h^0(C, H+2Z) = 2,
+ h^0(C, theta) = 3,
+ h^0(C, O_C(Z)) = 1,
+ h^0(C, theta + H) = 6,
+ H^0(C, L) = xi H^0(C, H),
+ H^0(C, H+2Z) = xi^2 H^0(C, H),
+ H^0(C, theta) = xi^2 H^0(C, 2H).
+In particular r = xi u is a basis of H^0(C, L) with common zero
+divisor exactly Z.
+
+(4) For every k >= 2, H^0(C, (k+1)L) -> H^0(Z, (k+1)L|_Z) is
+surjective.
+
+(5) The auxiliary data on Gamma (bundle identities, Omega, Q, and
+the six vectors w_j) satisfy w_j^2 + w_j . w = 1 and w^2 = 90 =
+9 deg L; and Z belongs to |M|.
+
+Proof. Throughout, C has bidegree (8,4) and K_C = (6,2)|_C. By
+intersection theory on the quadric,
+ deg H = (1,0).(8,4) = 4,
+ deg K_C = (6,2).(8,4) = 40,
+ g(C) = 21.
+Since deg Z = 6, deg L = 10. The morphism f given by u is the first
+projection restricted to the irreducible C, so it is finite of degree
+four. Tangency at q_j identifies the differential of f there with
+that of F|_Gamma, which is nonzero. The marked values f(q_j) equal
+the F|_Gamma-values at q_j, which are not all equal, so a regular
+fiber away from them exists. Replacing u by a determinant-one change
+of basis makes that fiber the zero divisor of u_1 and preserves
+eta_C and eta_Gamma.
+
+For 0 <= j <= 3, the defining-divisor sequence for C tensored with
+O_Sigma(j, 0) has kernel O_Sigma(j-8, -4), whose cohomology vanishes
+by product formula, so h^0(C, jH) = h^0(Sigma, O(j, 0)) = j + 1.
+
+For a line bundle B and effective divisor D on C, Riemann-Roch
+followed by Serre duality gives the dimension formula:
+ h^0(C, B+D) - h^0(C, B)
+ = deg D
+ - rank[H^0(C, K_C - B) -> H^0(D, (K_C - B)|_D)].
+
+The proof: tensor 0 -> O_C(-D) -> O_C -> O_D -> 0 by B, use
+h^1(C, B) = h^0(C, K_C - B)^*, and dualize the restriction map.
+
+Apply this with B = jH and D = 2Z (or D = Z when j = 0), and use the
+adjoint-jets ranks:
+ h^0(C, O_C(Z)) = h^0(C, O_C) + 6 - 6 = 1,
+ h^0(C, H+2Z) = h^0(C, H) + 12 - 12 = 2,
+ h^0(C, 2H+2Z) = h^0(C, 2H) + 12 - 12 = 3,
+ h^0(C, 3H+2Z) = h^0(C, 3H) + 12 - 10 = 6,
+using h^0(C, H) = 2 (from j = 1) and h^0(C, 2H) = 3 (from j = 2).
+The natural inclusions H^0(C, H) -> H^0(C, L) -> H^0(C, H+2Z)
+agree on dimensions 2, 2, 2, so both are isomorphisms. In particular
+H^0(C, L) = xi H^0(C, H), and h^0(C, L) = 2. Similarly H^0(C,
+theta) = xi^2 H^0(C, 2H), of dimension 3. The pencil u is base-point
+free by construction, so the common zeros of xi u equal div(xi) = Z.
+
+For (4), the defining-divisor sequence for N = (k+1)L followed by
+Serre duality shows that evaluation on Z is surjective if and only if
+the natural inclusion H^0(C, (3-k)L) -> H^0(C, (3-k)L + Z) is an
+isomorphism. For k = 2 this is L -> H+2Z, between spaces of dimension
+2; for k = 3 it is O_C -> O_C(Z), between spaces of dimension 1; for
+k >= 4 both spaces vanish because the target bundle has degree
+10(3-k) + 6 <= -4. Item (5) restates the six-vector lemma and the
+choice of Z. []
+
+Riemann-Roch and Serre duality give two further identities on C:
+ h^1(C, L) = h^0(C, 3L) = 12, h^0(C, 4L) = 21.
+Indeed, h^0(C, L) - h^1(C, L) = 10 + 1 - 21 = -10 and K_C - L = 3L,
+so 2 - h^0(C, 3L) = -10; and h^0(C, 4L) = h^0(C, K_C) = g(C) = 21.
+
+Finally, in compatible frames r = z u at each marked point, so
+r'(q_j) = u(q_j) up to the frame. The six rows r'(q_j) in C^2 span
+C^2: their projective directions are the marked values of f, and
+f(q_j) are not all equal.