editors: nobody
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).
pdf/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).
pdf/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).
pdf/greb-kebekus-kovacs-peternell-2011-1003.2913-diff-forms-lc.pdf
pdf/druel-1304.6000-log-canonical-spaces.pdf
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).
pdf/graf-1801.05753-low-genus.pdf
pdf/bergner-graf-2020-lipman-zariski-genus-one-higher.pdf
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.
pdf/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.
pdf/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.
pdf/serre-1956-gaga.pdf
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. These four points form two distinct reduced fibers of the Phi-map; the
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 following 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 - 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 following 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.
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.
The trace form uses Grothendieck's splitting theorem for vector
bundles on PP^1 (Amer. J. Math. 79, 1957, 121-138), and Grothendieck
duality as in Hartshorne, Residues and Duality, Lecture Notes in
Mathematics 20 (Springer 1966). It converts the quadratic-space data
on Gamma into a residue problem on C. Put theta = 2L,
Lambda = theta + H = 3H + 2Z, and let f: C -> PP^1 be the morphism
defined by u. With u the fixed basis of H^0(C, H), let
eta_C = u_1 du_2 - u_2 du_1 in H^0(C, K_C + 2H); the alternating
expression is independent of a frame choice for H. The analogous
eta_Gamma is already defined on Gamma.
Lemma (trace pairing). The rank-four vector bundle E = f_* Lambda on
PP^1 carries a perfect symmetric pairing
E tensor E -> O_{PP^1}, (T_1, T_2) |-> Tr_{C/PP^1}(T_1 T_2 / eta_C).
Its splitting type is (3, 0, 0, -3). Multiplication by xi^2 identifies
O_{PP^1}(3) with the positive line subbundle P of E. The form on global
sections has radical xi^2 H^0(C, 3H) and induces a nondegenerate
symmetric form <.,.> on
W = H^0(C, Lambda) / xi^2 H^0(C, 3H), dim W = 2.
For every x in PP^1, evaluation identifies this quadratic space with
(P^perp / P)_x.
Let q in Z be a marked point, with local frames fF of H, tau of Lambda,
and parameter z. Write, using the fixed isomorphism 2 Lambda =
K_C + 2H,
tau^2 = s(z) dz fF^2, eta_C = E(z) dz fF^2.
Coefficient evaluation ell_q : T |-> (T/tau)(q) descends to W, and in
the dual form on W^*,
ell_q^2 = E(q) / s(q).
In particular ell_q is nonzero and nonisotropic.
Proof. The morphism f is finite and flat of degree four. The identity
2 Lambda = K_C + 2H identifies Lambda^2 with the relative dualizing
bundle K_{C/PP^1} = K_C + 2H (equivalently, f^* K_{PP^1} = -2H).
Perfection is checked locally. At a ramification point choose a
parameter w on one component of f^{-1}(disk), with target coordinate
x = w^e. Choose a frame fF of H pulled back from the target disk so
that eta_C = dx fF^2. If tau^2 = a(w) dw fF^2 with a a unit, the local
pairing is (b, c) |-> Tr(a(w) b c / (e w^{e-1})). On the basis
1, w, ..., w^{e-1} the pairing without the unit a has matrix entries
Tr(w^{i+j} / (e w^{e-1})) = 1 if i + j = e - 1, else 0 (0 <= i,j < e).
The pairing is holomorphic and perfect; multiplication by the unit a
is an automorphism; taking the direct sum over the points above a
target disk gives perfection. On an unramified fiber the trace is the
sum of the four contributions, with no division by the degree.
The splitting type follows from Grothendieck's theorem. Self-duality
makes the splitting degrees pair as (a_1, a_2, -a_2, -a_1) with
a_1 >= a_2 >= 0. Proposition (curve data) gives
h^0(E(-1)) = h^0(C, Lambda - H) = h^0(C, theta) = 3,
h^0(E(-2)) = h^0(C, Lambda - 2H) = h^0(C, H + 2Z) = 2.
An O(a)-summand with a positive contributes a and a - 1 to these two
counts; all nonpositive summands contribute zero. Hence a_1 = 3 and
a_2 = 0, and E = O(3) + O^2 + O(-3).
The map O(3) -> E given by multiplication by xi^2 vanishes in the
lower-degree summands, so its image P is the positive summand, which
is a subbundle including over f(Z). The pullback
H^0(PP^1, O(3)) -> H^0(C, 3H) is injective between 4-dimensional
spaces and hence an isomorphism; thus H^0(P) = xi^2 H^0(C, 3H).
The line P is isotropic because Hom(O(6), O) = 0, and pairs trivially
with the two O-summads. Perfection gives P^perp = O(3) + O + O and
P^perp / P = O + O with a nondegenerate constant form. All global
sections of E lie in P^perp, and H^1(P) = 0. Taking sections proves
the assertions about W and its fibers.
At q in Z, coefficient evaluation in tau annihilates P_{f(q)} because
P is the image of xi^2. The fiber algebra over f(q) is Artinian, and
the trace pairing is the orthogonal sum of its factors. Since f is
unramified at q, the factor at q is one-dimensional with form
coefficient s(q) / E(q); evaluation at q vanishes on the other
factors. The metric dual of coefficient evaluation thus has square
E(q) / s(q) in the fiber E_{f(q)}; it lies in P^perp_{f(q)} because
evaluation kills P_{f(q)}; passing to the orthogonal quotient preserves
the square and gives the functional on W. Finally E(q) != 0 by the
unramifiedness of f at q. []
The dual form on W^* satisfies x^2 = x . x. The reciprocal in the
dual-evaluation identity comes from using this dual form rather than
the primal one.
Lemma (principal parts). There is an isomorphism
iota: W -> H^0(Gamma, Phi)
identifying the full principal parts of T/xi^2 on C with those of
iota([T])/beta^2 on Gamma through residue duality, and the projective
evaluations at corresponding points of Z. Its dual
iota^*: H^0(Gamma, Phi)^* -> W^* is an isometry for the quadratic
form Q and the dual trace form.
Proof. Principal parts are defined after fixing the logarithmic volume
Omega and its residue omega, so the residue convention is the one
introduced earlier. Set N_Sigma = Gamma + F, N_Gamma = N_Sigma|_Gamma,
and D = 2Z. Fourth-order contact identifies D on C and on Gamma as the
same embedded length-twelve subscheme of Sigma. The isomorphism
N_Sigma|_C = K_C - 3H is the one supplied by adjunction with the fixed
volume, and omega trivializes K_Gamma.
For a line bundle B on a smooth curve, the local pairing between
principal parts of B(D) and jets of K_C - B is
(a, s) |-> sum_{q in Z} Res_q(as).
It is perfect. In a local parameter with mutually dual frames, the two
coefficients of a_{-2} z^{-2} + a_{-1} z^{-1} pair with a jet
s_0 + s_1 z as a_{-2} s_1 + a_{-1} s_0.
The two principal-part spaces
H^0(C, O_C(3H + 2Z) / O_C(3H)),
H^0(Gamma, O_Gamma(-N_Gamma + 2Z) / O_Gamma(-N_Gamma))
are therefore both identified with H^0(D, N_Sigma|_D)^* by residues.
This defines their comparison without any choice of coordinates or
frames.
Both complete adjoint systems are restrictions of the same ambient
system H^0(Sigma, N_Sigma). The kernels in the restriction sequences
to C and to Gamma are respectively O_Sigma(-Gamma - 3F) = (-5, -2)
and O_Sigma(F) = (1, 0), with vanishing first cohomology for each.
Their images on the common scheme D therefore coincide. Equivalently,
an ambient section vanishing on Gamma vanishes to order at least four
on C at Z and has no effect on these residue tests.
The principal-part sequence together with Serre duality now says that
the image of H^0(C, Lambda) under T |-> pp(T/xi^2) is exactly the
annihilator of this common adjoint jet space, and its kernel is
xi^2 H^0(C, 3H). On Gamma the same annihilator is the image of
H^0(Gamma, Phi) -> H^0(Gamma, O_Gamma(-N_Gamma + 2Z) / O_Gamma(-N_Gamma)),
tilde(T) |-> pp(tilde(T)/beta^2),
because Phi = 2M - N_Gamma and div(beta) = Z. This map is injective:
a nonzero kernel element would be a nonzero section of -N_Gamma, but
deg(N_Gamma) = 10 > 0, so H^0(Gamma, -N_Gamma) = 0.
The two annihilators have the same dimension. On Gamma, h^0(N_Gamma)
= 10, and restriction to D is injective because
deg(N_Gamma - 2Z) = 10 - 12 = -2. So the kernel has dimension 10 and
the annihilator in the twelve-dimensional space has dimension 2. On
C the same dimension count gives dim W = 2. This gives the isomorphism
iota, and identifies it with principal parts.
For the metric, choose coordinates (y, z) near a marked point and
frames gm of O_Sigma(Gamma), fF of F, and m of M on Gamma with
gamma = y gm,
Omega = e(y, z) (dy/y) / dz, e(0, z) = 1,
kappa = k(y, z) gm^2 fF^4,
beta = b(z) m, m^4 = gm^2 fF^4|_Gamma,
so omega = dz on Gamma, with k(0, z) = b(z)^4 and
b'(0) k_y(0, 0) != 0. Put K(z) = k_y(0, z). Solving y = 0 on C gives
the contact expansion
y|_C = -b(z)^4 / K(z) + O(z^8),
e|_C = 1 + O(z^4),
k_y|_C = K(z) + O(z^4).
The normal-first adjunction sends the frame gm fF^4 to
Res_C( (e(y, z) dy / dz) / k(y, z) ) = (e|_C / k_y|_C) dz.
The sign is positive because dy / dz = (dk / k_y) / dz.
Let a = xi/z be the local frame of O_C(Z) and let tau = fF^3 a^2 be
the resulting frame of Lambda. The fixed fourth-power identification
sends a^4 to (y|_C / z^4) gm. Using the adjunction formula, its square
has the form
tau^2 = s(z) dz fF^2, s(z) = (e|_C y|_C) / (z^4 k_y|_C),
s(0) = -b'(0)^4 / k_y(0, 0)^2.
The negative sign comes from solving the equation of C.
Write T = hA(z) tau, and write tilde(T) = iota([T]) in the frame
tilde(tau) = m^2 / (gm fF) of Phi as tilde(T) = hAt(z) tilde(tau). For
an ambient test section S(y, z) gm fF, the two residue tests are
Res_0( S|_C hA(z) e|_C / (k_y|_C z^2) dz ),
Res_0( S(0, z) hAt(z) / b(z)^2 dz ).
The restrictions of S to the two curves agree modulo z^4. Equality of
their residue functionals on all local test jets is exactly
pp( hA / (K(z) z^2) ) = pp( hAt / b(z)^2 ),
hAt equals (b(z)/z)^2 / K(z) * hA mod z^2.
To exhibit both coefficients, put b = b_1 z + b_2 z^2 + O(z^3),
K = K_0 + K_1 z + O(z^2), hA = hA_0 + hA_1 z + O(z^2). The respective
principal parts are
hA_0 / K_0 z^{-2} + (hA_1 / K_0 - K_1 hA_0 / K_0^2) z^{-1},
hAt_0 / b_1^2 z^{-2} + (hAt_1 / b_1^2 - 2 b_2 hAt_0 / b_1^3) z^{-1}.
Hence
hAt_0 = (b_1^2 / K_0) hA_0,
hAt_1 = (b_1^2 / K_0) hA_1 + (2 b_1 b_2 / K_0 - b_1^2 K_1 / K_0^2) hA_0.
This invertible triangular transformation is the local expression of
the intrinsic residue-dual identification, so its expressions in
different coordinates and frames agree.
The metric agrees as follows. In these frames write
eta_C = E_C dz fF^2 and eta_Gamma = E_Gamma dz fF^2. Tangency gives
E_C(0) = E_Gamma(0). Let ell and tilde(ell) be coefficient
evaluations in tau and tilde(tau). The two-coefficient formula gives
iota^* tilde(ell) = alpha ell with alpha = b_1^2 / K_0. By the
dual-evaluation formula and the formula for s(0),
(iota^* tilde(ell))^2 = alpha^2 E_C(0) / s(0) = -E_Gamma(0).
The compatible bundle identification gives tilde(tau)^2 = fF^2, so
the last quantity is exactly Q(tilde(ell)), 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 vanishes identically. Hence iota^* is an isometry with sign
and scalar fixed. []
Proposition (residue data). The basis u can be chosen with unchanged
determinant so that (u_1)_0 is an unramified fiber of f disjoint from
Z. Set r = xi u and define v_j^* = iota^* w_j in W^*, and
V = sum_{j=1}^6 v_j^*. Then, for compatible coordinate disks at
q_1, ..., q_6, there exist nonzero constants c_j in C and a global
pair P = (P_1, P_2) in H^0(C, theta + Z)^2 with:
(1) c_j l_j = v_j^*, V^2 = 90 = 9 deg L, and
(v_j^*)^2 + v_j^* . V = 1.
(2) On the j-th disk, with coefficient rows in the indicated frames,
m_j = c_j / z_j, R_j = m_j r, W_j = R_j(0) / R_j'(0),
p_j = P - m_j r',
the pair P has principal part R_j(0) / z_j in the frame (l_j^fr)^2
of theta, both R_j and p_j are holomorphic near 0, and R_j(0) != 0.
(3) W_j = (v_j^*)^2, p_j(0) / R_j(0) = 1, and
sum_j (W_j - 1) = -9 deg L.
(4) J = u / P lies in H^0(C, Lambda) and, in W^*,
<[J], [T]> = -V([T]) for every T in H^0(C, Lambda).
(5) Changing P by a holomorphic pair of sections of theta leaves [J]
unchanged. The prescription is invariant under changes of compatible
coordinates.
Proof. A determinant-one change of basis preserves eta_C and
eta_Gamma and all forms already fixed. Avoiding the finitely many
branch values and the image of Z, one arranges that the zero divisor
of u_1 is the required fiber.
Fix on a small disk about q_j a parameter z_j and a frame l_j^fr of
L whose fourth power is dz_j (such a frame exists as a holomorphic
fourth root of a unit). Set
a_j = xi / z_j, fF_j = l_j^fr / a_j, tau_j = (l_j^fr)^2 fF_j.
These frame O_C(Z), H, and Lambda respectively. With r = xi u, the
coefficient row of r in l_j^fr is z_j u, where u denotes the
coefficient row in fF_j. Let t_j be the fiber-linear function on
L^{-1} corresponding to l_j^fr, giving compatible split coordinates
(t_j, z_j). Let l_j in W^* be coefficient evaluation in tau_j. Since
tau_j^2 = dz_j fF_j^2, the trace pairing gives
l_j^2 = (u_1 u_2' - u_2 u_1')(0).
Primes in the rest of this argument refer to z_j-derivatives.
The isometry iota transports the equations for w_j to
(vector-identities) and transports their evaluation directions. Since
each l_j is nonzero, there is a unique nonzero c_j with c_j l_j = v_j^*.
Locally R_j = c_j u, so R_j is holomorphic and nonzero at the center,
and the norm formula gives R_j(0) / R_j'(0) = c_j^2 l_j^2 = (v_j^*)^2.
Every prescribed set of simple principal parts of theta at Z is
realizable by a section of theta + Z. The obstruction in H^1(C, theta)
pairs, by Serre duality, with H^0(C, K_C - theta) = H^0(C, theta),
by the sum of residues. All these test sections vanish at Z (item (3)
of the curve-data proposition), so their residue against any simple
principal part is zero. Apply this separately to the two entries to
obtain P with the stated pole parts. Since r = z_j u, the row m_j r'
has exactly the same simple pole part as P, so p_j is holomorphic.
The simple pole of P at q_j is proportional to u(q_j). Its wedge with
u has zero residue coefficient, so J = u / P is holomorphic there;
away from Z it is already holomorphic. Hence J is a global section of
Lambda.
For the residue pairing, fix T in H^0(C, Lambda) and consider the
global meromorphic canonical differential
nu = P_1 T / u_1,
with theta + (theta + H) - H = K_C. Its only possible poles lie in Z
and in the simple zero fiber of u_1. At q_j the compatible frames make
the canonical product exactly coefficient multiplication followed by
dz_j, so Res_{q_j}(nu) = c_j l_j(T). At a zero q of u_1, the section
u_2 is nonzero, and
J(q) = -u_2(q) P_1(q), eta_C(q) = -u_2(q) du_1(q),
with the same minus sign in both expressions. In a local parameter,
Res_q(nu) = (JT/eta_C)(q).
The zero fiber is unramified, so the sum of these four residues equals
Tr(JT/eta_C) at that fiber. This trace is a global holomorphic
function on PP^1 (Lemma trace-pairing), hence constant. The residue
theorem gives
<[J], [T]> = -sum_j c_j l_j(T) = -V([T]),
which is (4).
On a fixed disk suppress the subscript j. Since r = z u,
P = p + (c/z) u + c u', c J = R / p + R / R'.
In particular c_j l_j(J) = R_j(0) / p_j(0) + W_j. On the other hand,
the metric identification in (4) gives c_j l_j(J) = -v_j^* . V =
W_j - 1. Hence R_j(0) / p_j(0) = -1, equivalently
p_j(0) / R_j(0) = 1. Summing W_j - 1 = -v_j^* . V proves the last
identity in (3), together with the local expansion
P = R_j(0) / z_j + p_j(0) + 2 R_j'(0) + O(z_j).
If P is changed by a holomorphic pair S, the equality
H^0(C, theta) = xi^2 H^0(C, 2H) shows u / S lies in
xi^2 H^0(C, 3H), so [J] is unchanged.
Compatibility with the fourth-root structure. Let zeta = psi(z) be
another centered parameter and g(z) l^fr the corresponding compatible
frame of L, with g^4 = psi'. The other frames change as
a_new = (z / psi(z)) a,
fF_new = g (psi(z) / z) fF,
tau_new = g^3 (psi(z) / z) tau.
With g_0 = g(0), l_new = g_0^{-7} l, so the same vector v^*
prescribes c_new = g_0^7 c. Also r_new = g^{-1} r and
R_new(0) = g_0^2 R(0). The coefficient row of the global theta-section
transforms as P_new = g^{-2} P. Its coefficient of zeta^{-1} is
psi'(0) g_0^{-2} R(0) = g_0^2 R(0), exactly the newly prescribed
R_new(0). Thus the same global P realizes the new prescription. The
row R_new is a scalar multiple of R with scalar g_0^2 at the center,
and differentiating with respect to zeta shows
W_new = g_0^4 W / psi'(0) = W. Finally, subtracting
m_new d r_new / d zeta from P_new shows p_new(0) - g_0^{-2} p(0) is
proportional to R(0), so p_new(0) / R_new(0) = p(0) / R(0). []
The finite-jet construction. On the total space of L^{-1} choose a base
coordinate z and a fiber-linear coordinate t, so that t^4 corresponds
to dz under the fixed isomorphism L^4 = K_C. Split changes take the
form
z -> zeta = psi(z), t -> t_hat = g(z) t, g^4 = psi'.
Put E_t = t d_t. The two-form
sigma = t^{-4} (dt/t) / dz = t^{-5} dt / dz
is invariant under such split changes, and has a pole of order five
along the zero section C = {t = 0}.
Seek vector fields of the form
v_i = t (A_i E_t + B_i d_z), i = 1, 2,
with Taylor series A_i = sum_{k >= 0} A_{ik} t^k, B_i = sum_{k >= 0} B_{ik} t^k
in t whose coefficients are holomorphic in z, and with initial data
A_0 = r = (r_1, r_2), B_0 = 0.
Then
sigma(v_1, v_2) = t^{-2} (A / B),
v_1 / v_2 = t^2 (A / B) E_t / d_z,
where A = (A_1, A_2) and B = (B_1, B_2) are two-entry rows.
The required logarithmic determinant is sigma(v_1, v_2) = 1 + O(t),
equivalently A / B = t^2 G with G a unit of the form 1 + O(t). Then
v_1 / v_2 = t^5 G d_t / d_z, so the ordinary determinant is t^5
times a nonzero unit along C.
Gluing. Around each of the six marked points choose a small disk D_j;
the six D_j are mutually disjoint. A neighborhood of C in the total
space of L^{-1} is described by a base chart U_0 over C minus the
union of the D_j and inside charts U_j over the D_j, glued along
annular overlaps by transition maps. The construction proceeds by
modifying the transition maps through finitely many holomorphic
parameter families. Admissible modifying maps are time-one flows of
holomorphic vector fields
D = t^k (b(z) E_t + h(z) d_z), k >= 1,
on the annular overlaps. Using the flow of -D, the induced action on
other vector fields is exp(ad D). Every generator vanishes on the
zero section and changes its local equation only by t |-> t + O(t^2),
so the zero section and its conormal bundle are preserved.
Action on coefficients. For D as above and V = t^{l+1} (A E_t + B d_z),
[D, V] = t^{k+l+1} ( ((l+1-k) b A + h A' - B b') E_t
+ ((l+1) b B + h B' - k h A - B h') d_z ).
The logarithmic divergence is
L_D sigma = t^k ((k - 4) b + h') sigma.
For k != 4, choosing b = h' / (4 - k) preserves sigma exactly. On the
leading row of index 0, the effect reduces to
Delta A_k = (1 - k) b r + h r',
Delta B_k = -k h r.
Concretely, a modifying field of leading index k acts on the row
indexed k by this rule, and on row indices greater than k by
corrections that appear at strictly higher t-orders.
Existence of flows. On a compact sub-annulus every D in the above
form is holomorphic near the zero section and vanishes there. So its
time-one flow and inverse exist after shrinking the fiber disc once
and for all six disks.
Order one. On the j-th overlap, take h = m_j and b = m_j'/3. Choose
(A_1, B_1) = (p, m_j r) on U_j,
(A_1, B_1) = (P, 0) on U_0.
Lemma (bracket) with k = 1, A = r, B = 0 gives Delta A_1 = m_j r' and
Delta B_1 = -m_j r. Since P = p + m_j r' by the residue-data
proposition, the first transition changes the inside leading row to
(P, 0), exactly matching the outside leading row.
The degree-one coefficient of A / B is A_0 / B_1 + A_1 / B_0.
On the outside it is r / 0 + 0 / P = 0. On the inside it is
r / (m_j r) + p / 0 = 0, since r / r = 0. So no obstruction
arises at this order.
Order two. The order-one transition acts on the row of index 2 via
exp(ad D_1), D_1 = t (m' E_t/3 + m d_z), with m = m_j. Expanding in
the two components gives
Delta A_2 = (1/3) m' p + m p' + (2/3) m m' r' + (1/2) m^2 r''
- (1/6) m m'' r,
Delta U = -m p + (1/3) m m' r.
Substituting m = c/z, so R = m r, R' = mr' + m' r, and using
r = z R / c, gives the Laurent expansions
Delta A_2 = -(c R)/z^3 + (c R')/(3 z^2) + (c R'')/(2 z)
- (c p)/(3 z^2) + (c p')/z,
Delta U = -(c p)/z - (c R)/(3 z^2).
Define, using z = z_j and c = c_j on the j-th disk,
F_ij = Res_0(r_i Delta A_2, j dz),
H_ij = Res_0(r_i' Delta U_j dz).
Direct computation from the Laurent expansions gives
F_ij = -R_i p_j / 3 - (2/3) R_i R_j' - R_i' R_j,
H_ij = -R_i p_j - R_i R_j' / 3 - (2/3) R_i' R_j.
The order-two obstruction is the residue matrix
T_ij = F_ij + H_ij/4 + 3 H_ji/4
= -7 R_i p_j / 12 - 9 R_j p_i / 12
- 5 R_i R_j' / 4 - 17 R_i' R_j / 12.
Decompose T_ij into symmetric and antisymmetric parts:
T^sym_ij = -(2/3)(R_i p_j + R_j p_i + 2 R_i R_j' + 2 R_i' R_j),
T^alt_ij = (1/12) epsilon_ij ((R / p)(0) + (R / R')(0)).
For the symmetric part, P_i P_j is a meromorphic section of
L^4 = K_C with local expression P_i P_j dz. Using the expansion
P = R(0)/z + p(0) + 2 R'(0) + O(z) from the residue-data proposition,
Res_0(P_i P_j dz) = R_i p_j + R_j p_i + 2 R_i R_j' + 2 R_i' R_j.
Summing over j and applying the residue theorem on C shows that
sum_j T^sym_ij = 0.
For the antisymmetric part, note R(0) / p(0) = -1, and
sum_nu (W_nu - 1) = -9 deg L. So
sum_j T^alt_ij = (1/12) epsilon_ij sum_nu (W_nu - 1)
= -(3/4) epsilon_ij deg L.
Hence the four column sums of T_ij in the two indices, i.e., the four
totals of the entries (11), (12), (21), (22), equal
0, -(3/4) deg L, (3/4) deg L, 0.
By the residue-data proposition, both the inside and outside data
evaluate the same principal-part pairing V, so these totals are also
the residue totals computed from the outside chart. Consequently the
vertical difference lies in H^1(C, L^3)^2 and pairs trivially, by
Serre duality, with every element of H^0(C, K_C - 2L) = H^0(C, 2L) =
H^0(C, theta). The class in H^1(C, L^3)^2 is therefore zero, and
holomorphic A_2, B_2 on the two charts solve the matching.
Higher orders. From order three onward, each step extends fields
together with the required determinant. Write the
determinant equation as
A / B = t^2 G, G = 1 + G_1 t + G_2 t^2 + ...
The field equation in degree k uses G_{k-2}. To compare the next
horizontal rows, that target coefficient must already be compatible,
so each target coefficient is chosen before the field equation it
enters is imposed.
Local bivector target. On a neighborhood Y of C, put I = O_Y(-C) and
F = T_Y(-log C). The relevant line bundle is
M = I^4 det F = I^5 det T_Y, M|_C = L^4 tensor T_C = O_C,
using 4L = K_C and T_C = K_C^{-1}. Its local frame is sigma^{-1}; the
degree-k associated graded piece is L^k. Transport under an order-k
transition D = t^k(b E_t + h d_z) with divergence
q = div_sigma D = (k-4)b + h' is
[D, G sigma^{-1}] = (D(G) - qG) sigma^{-1},
G |-> exp(D - q) G,
where D - q acts by f |-> D(f) - qf. Every order-k transition changes
a target with constant term 1 first in degree k, by -((k-4)b + h'). A
leading degree-k discrepancy is a cocycle in L^k.
Choice ordering. For field changes at step k, the first determinant
degrees affected at each step are
- global addition to A_{k-1}: degree k (target G_{k-2})
- B_k: degree k (target G_{k-2})
- A_k: degree k+1 (target G_{k-1}).
The target coefficients in the last column stay fixed during these
changes. So A_{k-1} can be adjusted at degree k without disturbing
determinant equations imposed earlier. G_k is chosen after fields
match through index k, since its field equation occurs in degree k+2.
Invariant. After step n >= 2: rows through index n match; A/B is
imposed through degree n; G_0 = 1, ..., G_n are chosen and compatible
through degree n. For n = 2 both transitions preserve sigma exactly,
so the invariant holds with G = 1.
Step k >= 3. Assume the invariant through step k-1. On each chart,
r / B_k = G_{k-2} - sum_{a=1}^{k-1} A_a / B_{k-a}, (in L^{k-2}).
Under split changes, the only extra terms are multiples of the
antisymmetric sum sum_{a=1}^{k-1} B_a / B_{k-a} = 0.
Making the RHS divisible at Z. Adding any a in H^0(C, L^k)^2 to the
previously chosen A_{k-1} preserves the matching through index k-1 and
all previous determinants: every previous nonsplit transition has
positive order, and its first determinant contribution is a / B_1 in
L^k. Item (4) of the curve-data proposition (with k >= 3) gives a
surjection H^0(C, L^k) -> H^0(Z, L^k|_Z), so a(q_j) may take any
prescribed values. Since B_1(q_j) = R_j(0) != 0, the scalar map
a(q_j) |-> a(q_j) / B_1(q_j) is onto, so a can be chosen so that
(RHS - a / B_1)(q_j) = 0 for every j. Then the RHS is divisible by
r. Since r = z u has only a simple common zero, dividing by z on
each disk and applying Steinness on U_0 gives a holomorphic B_k.
Previous transitions act on the new B_k by
delta A_k = ((k-1)/3) m' a + m a',
delta B_k = -m a.
The horizontal correction satisfies r / (-m a) = a / (m r),
matching the determinant change on the disk. These contributions
are absorbed into the comparisons below.
Matching horizontal rows. Choose holomorphic vertical lifts of the
new B_k. After transport by previous transitions, all lower rows
agree. The difference of the degree-k A/B is r / delta B_k
(because delta A_k / B_0 = 0), and target compatibility through
degree k-2 makes it zero. The horizontal discrepancy is thus a
holomorphic multiple of r on each overlap, and (1-k)b r + h r'
removes it by choice of h in an order-k transition. Take b = h' when
k = 3 and b = 0 when k >= 4. The new transition first affects G in
degree k, so it cannot disturb the target comparison of degree k-2
just used.
Matching vertical rows. After horizontal matching, the remaining
class lies in H^1(C, L^{k+1})^2. For k >= 4 it vanishes, since
H^1(C, L^{k+1}) = H^0(C, L^{3-k})^* = 0. Holomorphic vertical
corrections on the charts solve it without touching the current
determinant equation, because B_0 = 0.
At k = 3 the obstruction lies in H^1(C, K_C)^2 = C^2, tested by
constants in the two entries. The added freedom is: on the j-th
overlap replace h by h + lambda_j/z, and on its disk compensate by
B_3 |-> B_3 + 3 lambda_j r / z.
The compensation is holomorphic and wedges to zero with r. It
preserves the current determinant equation and horizontal matching:
the transition adds the opposite horizontal row -3 lambda_j r / z.
Keep the vertical lift A_3 unchanged in the chosen disk coordinate.
Changing a degree-three input row affects its transport under
previous positive-order transitions only at higher indices. The
change in the transported vertical difference at index three is
lambda_j (r'/z + 2 r/z^2),
whose residue at 0 is 3 lambda_j r'(0). The two entries are
coefficients of canonical differentials, since L^4 = K_C. The six
rows r'(q_j) span C^2 by the curve-data proposition, so the map
(lambda_1, ..., lambda_6) |-> sum_j 3 lambda_j r'(q_j) has rank two
and cancels both entries. The resulting vertical class splits,
completing order three. The modified order-three transition still
has b = h', so it preserves sigma exactly.
The correction can affect the degree-four coefficient of A/B.
No conflict arises: G_2 = 0 is already chosen, but its first
determinant equation is at step four; the degree-three equation is
unchanged.
Extending the target. After the order-k transition, choose G_k. For
k = 3 all transitions preserve sigma, so G = 1 continues to work. At
k = 4 the earlier transitions still preserve sigma, so the first
target discrepancy is -h' in degree four, by the target transport
formula with b = 0. The horizontal coefficient of an order-four
generator has type L^4 tensor T_C = O_C; its split transformation
h -> g^{4-k} h equals h at k = 4. Thus h' dz = dh is an exact
differential on the overlap, with zero contour residue. Its sole
obstruction to splitting is the pairing with the constant section of
O_C, so it vanishes. There are holomorphic chart coefficients G_4
with G_1 = G_2 = G_3 = 0. Transport by a previous positive-order
modification changes a degree-four term only in degrees at least
five, so the residue argument applies at degree four.
For k > 4 the new leading discrepancy lies in L^k, and
H^1(C, L^k) = H^0(C, L^{4-k})^* = 0
because 4-k < 0. Thus it always splits and defines G_k. Every target
extension leaves all previously imposed equations unchanged, since
G_k enters A/B only in degree k+2. The induction invariant is
preserved.
Stopping. A row of index k contributes to degree k+1 of the
logarithmic frame (E_t, d_z), so matching through index 7 prescribes a
section modulo I^9 F.
Proposition (finite compatible jets). There exist finitely many
holomorphic disk-to-outside transition germs, each a finite
composition of flows of generating fields D = t^k(b E_t + h d_z) for
1 <= k <= 7, together with holomorphic local rows A_k, B_k for
0 <= k <= 7, such that: the transitions fix C and preserve its
conormal bundle L; the vector fields v_i = t(A_i E_t + B_i d_z)
match through row index 7; and A / B = t^2(1 + O(t)) on every chart.
On any surface neighborhood Y realizing these germs, with
I = O_Y(-C) and F = T_Y(-log C), the rows give two sections of
F/I^9 F inside I F/I^9 F. The determinant has leading term
t^4 E_t / d_z, equivalently t^5 d_t / d_z, with coefficient one.
Proof. Perform the first two orders and the induction through
k = 7. Only finitely many flows occur, each a genuine holomorphic
germ on an annular neighborhood of the zero section. Terms with row
indices 0, ..., 7 have ordinary-degree coefficients 1, ..., 8 in the
frame (d_t, d_z), so matching them is matching modulo I^9 F. The
determinant equation with G = 1 + O(t) gives
v_1 / v_2 = t^4(1 + O(t)) E_t / d_z = t^5(1 + O(t)) d_t / d_z. []
Analytic realization and Grauert contraction. The finite jets live
in overlapping charts, and are realized
on a single Hausdorff complex surface Y containing C; the vector
fields are then lifted from their jets by a cohomology vanishing
argument; and the curve C is contracted to a singular point.
Throughout, powers of an ideal denote ordinary ideal
powers, and cohomology is analytic sheaf cohomology.
Coherence of analytic structure and ideal sheaves, and the Hartogs
extension property for normal surfaces, are taken from Grauert and
Remmert, Coherent Analytic Sheaves, Grundlehren 265 (Springer 1984),
and from Demailly, Complex Analytic and Differential Geometry (2012).
Lemma (Hausdorff neighborhood). The overlap maps of the finite-jets
proposition define, after shrinking their domains, a smooth Hausdorff
complex surface Y containing C. Its conormal bundle is L, and its
self-intersection is C^2 = -10. Writing I = O_Y(-C) and
F = T_Y(-log C), the prescribed rows define two global sections
v_1bar, v_2bar in H^0(Y, F / I^9 F).
In divisor coordinates (t, z), their local representatives have
logarithmic determinant t^4(1 + O(t)) and ordinary determinant
t^5(1 + O(t)).
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 ( b(z) t d_t + h(z) d_z ), k >= 1,
on punctured disk neighborhoods. On a compact annulus in any such
overlap, these fields vanish along the zero section. The local
existence theorem for holomorphic differential equations, applied
uniformly on that compact annulus, gives the required time-one maps
and their inverses after shrinking the fiber radius. There are only
finitely many flows. Shrinking successively gives their compositions
on neighborhoods of the same compact annular bands. Denote the
resulting inside-to-outside maps by Phi_j for 1 <= j <= 6.
A quotient construction also verifies the separation
property. Choose mutually disjoint coordinate disks D_j about the
six points, and in each radii
0 < a_j < a'_j < b'_j < b_j
so that Phi_j and its inverse are defined on neighborhoods of the
closed annulus a_j <= |z_j| <= b_j in their respective split
bundles. Their domains may be taken somewhat wider than the closed
annuli. Retain a small open neighborhood V_j of the zero section
over |z_j| < b'_j, and a small open neighborhood V_0 in N over
C minus the union of {|z_j| <= a'_j} for 1 <= j <= 6. Choose the
transverse thicknesses small enough that the maps are defined
throughout the retained parts of the closed annuli and
|z_j(Phi_j(p)) - z_j(p)| < (a'_j - a_j) / 2
there. This is possible because each map restricts to the identity
on the zero section. Make the images of the retained annuli lie over
their respective disjoint disks 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. A point of V_0 lies in at most
one of their images, since those images lie over disjoint disks.
Each equivalence class in the disjoint union V_0 union V_1 union
... union V_6 thus has at most two elements. In particular the
equivalence relation consists of the diagonal, these six graphs,
and their inverses; no further identifications or triple-overlap
equations occur.
Each graph is closed relative to V_j x V_0. Indeed, suppose
p_nu in A_j with p_nu -> p in V_j and Phi_j(p_nu) -> q in V_0. The
source base radii have limit at least a_j and less than
b'_j < b_j. The flow is defined across the closed annulus, so
continuity gives q = Phi_j(p). If the limiting source radius were
a_j, the collar ineuality would give |z_j(q)| < a'_j, contrary to
q in V_0. Thus a_j < |z_j(p)| < b_j and p in A_j, as required.
Possible fiber boundaries cause no difficulty because both limiting
points are assumed to belong to the retained open pieces.
The equivalence relation is therefore closed. The quotient map is
open, since it glues open sets by biholomorphisms. An open quotient
map with closed equivalence relation has Hausdorff target: two
ineuivalent representatives have product neighborhoods disjoint
from the relation, and the images of these neighborhoods are
disjoint open sets. Moreover, the quotient map restricts on every
V_i to an injective open map. The complex structures on these
pieces consequently give the quotient a smooth complex surface
structure. The images of their zero sections give the original
compact curve C with its original complex structure.
Every modifying flow fixes C and induces the identity on its
conormal bundle. In the split identification its coordinate
expansion has t' = t + O(t^2), although its base coordinate may
change by O(t). Composing with the original split changes therefore
leaves I / I^2 = L. This proves C^2 = -deg L = -10.
In the logarithmic basis (t d_t, d_z) the prescribed fields have
coefficients t A_i, t B_i. Rows with indices 0 through 7 prescribe
these coefficients through degree 8 in t. The matching in the
finite-jets proposition is therefore precisely matching modulo
I^9 F on the actual overlap maps, giving the claimed quotient
sections. Their logarithmic determinant is
t^2 (A / B) = t^4(1 + O(t)). Replacing t d_t by d_t multiplies
this determinant by t, giving the ordinary determinant. []
Vanishing and holomorphic lifting. The finite jets now live on a
Hausdorff surface. To lift them to holomorphic fields, the first
cohomology of I^9 F vanishes after shrinking about C. The
logarithmic tangent sheaf F is locally free of rank two.
Restriction to C gives the exact sequence
0 -> O_C -> F|_C -> T_C -> 0.
The first map sends 1 to the class of the normal Euler field
t d_t, and the second is restriction of a tangent vector field to C.
The Euler class is unchanged when the local equation of C is
multiplied by a unit. The sequence is intrinsic; a splitting is
unnecessary.
Lemma (graded vanishing). For every n >= 9,
H^1(C, F|_C tensor L^n) = 0.
Consequently, with E = I^9 F and every r >= 1,
H^1(Y, E / I^r E) = 0.
Proof. Since K_C = L^4, tensoring the log sequence by L^n gives
0 -> L^n -> F|_C tensor L^n -> L^{n-4} -> 0.
Serre duality identifies the duals of the first cohomology groups of
the outer terms with
H^0(C, L^{4-n}) and H^0(C, L^{8-n}),
respectively. Both vanish for n >= 9, since deg L > 0. The first
assertion follows from the cohomology sequence.
Because C is a Cartier divisor and F is locally free,
I^n F / I^{n+1} F = F|_C tensor L^n.
The quotient E / I^r E = I^9 F / I^{9+r} F has a finite filtration
with precisely these graded terms for 9 <= n < 9+r. Their first
cohomology vanishes by the first assertion. Successive cohomology
sequences give the second assertion; the quotient is supported on
the compact curve C. []
Lemma (actual lifting). After shrinking Y about C, the sections
v_1bar, v_2bar lift to holomorphic sections v_1, v_2 of F, producing
a frame for T_{Y minus C}.
Proof. The self-intersection matrix is the negative definite
(-10). Grauert's contraction criterion therefore contracts C to a
point x of a normal complex surface:
pi: Y -> X, pi^{-1}(x) = C,
Y minus C is isomorphic to X minus {x},
with pi_* O_Y = O_X. The criterion is stated in Grauert, "Ueber
Modifikationen und exzeptionelle analytische Mengen", Math. Ann. 146
(1962), 331-368. The contraction is proper and is a biholomorphism
off the exceptional curve; it is the holomorphic reduction in that
theorem, whose target is normal when the source is normal. Choose a
sufficiently small Stein representative X about x and replace Y by
its inverse image; then X minus {x} is smooth.
Vanishing of R^1 pi_* E. Put E = I^9 F, and let m_x be the ideal
sheaf of x in X, and J = m_x O_Y. Both J and I vanish exactly on C,
and I is radical. The analytic Nullstellensatz and Noetherianity
give an inclusion I^a subset J locally at each point of C for some
positive integer a. A finite cover of the compact curve gives a
single exponent, and shrinking the representative makes
I^a subset J subset I,
I^{ar} subset J^r subset I^r for r >= 1.
The proper analytic direct-image theorem makes R^1 pi_* E coherent.
The theorem on formal functions (Grauert, Publ. Math. IHES 5, 1960,
5-64) identifies its completed stalk with
(R^1 pi_* E)_x completed = inverse limit over r of H^1(Y, E / J^r E).
The inclusions above give J^{ar} E subset I^{ar} E subset J^r E. So
the transition from index ar to index r factors through
H^1(Y, E / I^{ar} E) = 0 by graded vanishing. The inverse system is
therefore pro-zero, and its limit is zero. The stalk (R^1 pi_* E)_x
is a finite module over the Noetherian local ring O_{X, x}, so its
zero completion implies that the stalk itself is zero, by Nakayama's
lemma. Away from x the map pi is an isomorphism, so the same sheaf
vanishes there. Thus R^1 pi_* E = 0.
pdf/grauert-1960-theorem-garbentheorie.pdf
The sheaf pi_* E is also coherent. Cartan's theorem B on the Stein
space X, plus the Leray spectral sequence for pi, gives
H^1(Y, E) = 0. The cohomology sequence of
0 -> E -> F -> F / I^9 F -> 0
lifts v_1bar, v_2bar to holomorphic sections v_i of F.
Any such lift still frames. In each divisor chart, let v_i^{(8)} be
the polynomial representative of the prescribed jet. It belongs to
I F, and v_i - v_i^{(8)} belongs to I^9 F. So the change in their
wedge lies in I^{10} det F. The logarithmic determinant of the
lifted pair is still t^4(1 + O(t)); its leading coefficient is 1
even at the common zeros of the leading row r. Its ordinary
determinant is
v_1 / v_2 = t^5 u(t, z) d_t / d_z, u(0, z) = 1.
The unit condition holds on a neighborhood of each point of C.
Compactness gives one neighborhood of C where all local factors are
nonvanishing, and properness allows a further shrinking of X whose
inverse image lies in this neighborhood. So v_1, v_2 frame
T_{Y minus C}. []
Proposition (counterexample). There is a normal complex surface germ
(X, x) with an isolated singular point and free analytic derivation
sheaf of rank two:
T_X = Hom_{O_X}(Omega^1_X, O_X) = O_X^2.
It has a resolution pi: Y -> X whose exceptional set is the smooth
curve C of genus 21 and self-intersection -10.
Proof. Take the contraction constructed in the lifting lemma.
Suppose x were a smooth point. A sufficiently small neighborhood of
x admits a nowhere-vanishing holomorphic two-form omega. Its
pullback is holomorphic on Y and nonvanishing off C. Thus
div(pi^* omega) = a C with a >= 0,
K_Y . C = a C^2 <= 0.
Adjunction, however, gives
K_Y . C = 2 g(C) - 2 - C^2 = 40 + 10 = 50.
The contradiction shows x is singular, and it is the only singular
point after the chosen shrinking.
Write X^o = X minus {x}, and j: X^o -> X. The two fields of the
lifting lemma descend to a holomorphic frame w_1, w_2 on X^o.
Normality in dimension two gives the Hartogs extension
O_X = j_* O_{X^o}.
This property extends the fields as derivations, as follows. Embed a
representative of (X, x) in a smooth analytic germ with coordinate
functions x_1, ..., x_e. For each i and alpha, the holomorphic
function w_i(x_alpha) on X^o extends uniquely to a function a_{i
alpha} on X. If f is any local defining equation, then
sum_alpha a_{i alpha} df/dx_alpha = 0 on X^o and hence on X, by
density and reducedness. The tuple (a_{i alpha}) defines a section of
Hom_{O_X}(Omega^1_X, O_X) extending w_i, still denoted w_i.
For any local analytic derivation w on X, the frame on X^o gives
unique holomorphic coefficients h_1, h_2 there with w = h_1 w_1 +
h_2 w_2. Both coefficients extend by Hartogs. The extended equality
holds on X, since applying its difference to embedding coordinates
gives holomorphic functions vanishing on the dense open set X^o.
This proves surjectivity of
O_X^2 -> T_X, (h_1, h_2) |-> h_1 w_1 + h_2 w_2.
If a pair lies in the kernel, both coefficients vanish on X^o because
w_1, w_2 are a frame there, and therefore vanish on X. The map is
also injective, proving the assertion. []
Invariants. The construction also determines the canonical
discrepancy and a lower bound for the geometric genus. The two-form
omega^o dual to the frame w_1, w_2 generates the canonical sheaf on
X^o. Extending reflexively across the normal surface point gives
omega_X = O_X; equivalently, K_X = 0. A normal surface is
Cohen-Macaulay, so this reflexive rank-one sheaf is its dualizing
sheaf, and X is Gorenstein; see Flenner and Zaidenberg, "Rational
curves and rational singularities" (2001; arXiv:math/0109221), and
Serre, "Un theoreme de dualite", Comment. Math. Helv. 29 (1955),
9-26.
pdf/flenner-zaidenberg-2001-math0109221-rational-curves-singularities.pdf
By the ordinary determinant formula, the pullback of omega^o is
pi^* omega^o = t^{-5} u(t, z)^{-1} d_t / d_z.
It is meromorphic on Y with divisor exactly -5C. Thus
K_Y = pi^* K_X - 5C,
so the discrepancy of C is -5.
For the geometric genus, put O_{nC} = O_Y / I^n. Riemann-Roch and
the established equality h^0(C, L) = 2 give
h^1(C, L) = h^0(C, L^3) = 12.
In the sequence
0 -> L -> O_{2C} -> O_C -> 0,
the constant section of O_C lifts to O_{2C}. Since H^2(C, L) = 0,
its cohomology sequence gives
h^1(O_{2C}) = h^1(C, L) + h^1(C, O_C) = 12 + 21 = 33.
For every n >= 1, the sequence with kernel L^n shows that
H^1(O_{(n+1)C}) -> H^1(O_{nC}) is surjective. The theorem on formal
functions with the cofinal filtrations above identifies their inverse
limit with the completion of (R^1 pi_* O_Y)_x. This stalk has finite
length, since the coherent higher direct image is supported at x,
and thus equals its completion. The inverse limit maps onto
H^1(O_{2C}), proving
p_g(X, x) = dim (R^1 pi_* O_Y)_x >= 33.
The exceptional curve has genus g = 21 and its dual graph has first
Betti number b = 0. Consequently p_g - g - b >= 12.
Algebraization. The passage from the analytic germ to an affine
variety approximates the defining equations together with a finite
relation matrix; their exact zero product and the closeness of the
approximation preserve the complete local ring. Tangent freeness
then descends from the continuous derivations of that ring. This
follows the equation-and-relation determinacy method of Hironaka,
"Formal line bundles along exceptional loci" (Tata Inst. Fund. Res.
Stud. Math. 4, 1969, 201-218), used in Artin's isolated-singularity
algebraization, "Algebraic approximation of structures over complete
local rings", Publ. Math. IHES 36 (1969), 23-58. The finite-
determinacy statement is proved below, with fixed estimates and no
complete-intersection hypothesis.
pdf/artin-1969-algebraic-approximation.pdf
Put P = C[[x_1, ..., x_e]], n = (x_1, ..., x_e), I = (f_1, ...,
f_s), R = P / I, where f = (f_1, ..., f_s) in n P^s is a row.
Write m_R = n R, and let an overbar denote reduction modulo I. Choose
an s x t matrix D whose columns generate all relations on f. Thus
P^t -> P^s -> I -> 0
is exact, with the first map given by multiplication by D and the
second by f. Define
psi: R^s -> R^t, v |-> v Dbar,
K = ker psi,
J = sum_{j=1}^e R * (d_j f-bar).
Differentiating f D = 0 and reducing modulo I gives J subset K.
Right-exact reduction of the presentation gives a presentation of
I/I^2; its dual identifies
K = Hom_R(I/I^2, R), T^1 = K/J.
In particular, no exactness on the left after reduction is assumed.
Lemma (finite determinacy with a relation matrix). Suppose T^1 has
finite length. There exist integers N, n_0 depending only on f, D
such that the following holds. If a row g in P^s and an s x t
matrix B satisfy
gB = 0,
g - f in n^{n_0} P^s,
B - D in n^N P^{s x t},
then there exist a continuous C-algebra automorphism Phi of P and an
invertible matrix U in GL_s(P) such that
f = Phi(g) U.
One can take Phi(x_j) = x_j mod n^2 and U = 1 mod n. Consequently
P/(g) = P/(f) as complete local C-algebras.
Proof. The equations are improved one order at a time. The exact
relation gB = 0 makes the equation error, reduced modulo I, lie
nearly in K. Artin-Rees corrects it into K; the finite length of
K/J then expresses the corrected class by coordinate derivatives. A
change of generators removes the remaining I-valued error. The
corrections are arranged so equation precision increases at every
step while relation precision stays fixed. The coordinate and
generator changes then converge in P.
Bounds. For a map u: M -> M' of finite modules over a Noetherian
local ring with maximal ideal a, Artin-Rees applied to im(u) in M'
gives a constant alpha(u) >= 0 such that
im(u) intersect a^r M' is contained in u(a^{r - alpha(u)} M)
for r >= alpha(u). Choose alpha, beta, d >= 0 so that
im(psi) intersect m_R^r R^t is contained in psi(m_R^{r - alpha} R^s),
K intersect m_R^r R^s is contained in m_R^{r - beta} K,
I^{⊕ s} intersect n^r P^s is contained in
{ fC : C in n^{r - d} P^{s x t} },
for all relevant r. In the last line, Artin-Rees applies to the map
C |-> fC: P^{s x t} -> P^s with image I^{⊕ s}; the relevant
filtration is the one induced from P^s.
Choose l >= 0 with m_R^l K contained in J, and set
c_0 = beta + l, c = c_0 + d,
N >= alpha + 1, n_0 >= max{c + N, c + 2, 2c + 1}.
Improvement step. Suppose n >= n_0 and
h = g - f in n^n P^s, B - D in n^N P^{s x t}, gB = 0.
Reduced modulo I, the exact relation and the closeness give
hbar Dbar = gbar Dbar = -gbar (Bbar - Dbar) = -hbar (Bbar - Dbar)
in m_R^{n+N} R^t. The factor hbar on the right provides order n + N.
Artin-Rees on psi gives epsilon-bar in m_R^{n+N-alpha}, contained in
m_R^{n+1}, with psi(epsilon-bar) = psi(hbar). Then
kappa = hbar - epsilon-bar in K intersect m_R^n
is contained in m_R^{n-beta} K
is contained in m_R^{n-c_0} J,
where the last inclusion uses m_R^l K contained in J with remaining
factor n - beta - l = n - c_0. Lift coefficients of kappa in the
generators d_j f-bar to elements a_j in n^{n-c_0}, and lift
epsilon-bar to epsilon in n^{n+1} P^s. Then
w = h - sum_j a_j d_j f - epsilon
lies in I^{⊕ s} intersect n^{n-c_0} P^s. The order of this
ideal-valued remainder need only be n - c_0, since a derivative of a
defining equation can be a unit. Artin-Rees on the ideal gives
w = fC with C in n^{n-c} P^{s x t}. So
h = sum_j a_j d_j f + fC + epsilon,
a_j, C in n^{n-c}, epsilon in n^{n+1} P^s.
The loss d for lifting the ideal-valued remainder is included in the
fixed constant c.
Correction. Put q = n - c, and define
phi(x_j) = x_j - a_j, M = (1+C)^{-1},
g_new = phi(g) M, B_new = (1+C) phi(B).
The bounds give q >= N, 2q >= n + 1, n + q - 1 >= n + 1. In
particular phi is a formal automorphism with identity linear part,
and M is invertible. Formal Taylor expansion gives
phi(f) - f + sum_j a_j d_j f in n^{2q} P^s,
phi(h) - h in n^{n+q-1} P^s.
Combined with the linear correction,
phi(g) = f(1+C) mod n^{n+1} P^s,
g_new - f in n^{n+1} P^s.
Also g_new B_new = phi(g)(1+C)^{-1}(1+C) phi(B) = 0 exactly. And
B_new - D = C phi(B) + (phi(B) - B) + (B - D) in n^N P^{s x t},
since C has order at least q >= N and substitution by phi changes
a power series by an element of n^q. The relation precision N is
preserved while the equation precision increases from n to n+1.
Iteration. Starting from (g_{n_0}, B_{n_0}) = (g, B), repeat for
n = n_0, n_0+1, ... with cumulative corrections
Phi_{n_0} = id, U_{n_0} = 1,
Phi_{n+1} = phi_n o Phi_n,
U_{n+1} = phi_n(U_n) M_n,
so that g_n = Phi_n(g) U_n at every stage. Since
phi_n - id has coefficients in n^{n-c} and M_n - 1 has entries in
n^{n-c}, the consecutive images Phi_n(x_j) and matrices U_n differ
in orders tending to infinity, and converge in the complete ring P
to Phi(x_j) and U. The resulting substitution Phi has identity
linear part and hence is an automorphism, while U = 1 mod n is
invertible. Taking limits in g_n = Phi_n(g) U_n gives
f = Phi(g) U. []
The original D must generate all relations, in order that
K = Hom_R(I/I^2, R) holds. No such assertion is needed for the
approximating matrix B: its exact zero product and its closeness to
D are sufficient. The lemma therefore concerns solutions of the
finite system gB = 0, not arbitrary truncations of a list of
defining equations.
Proposition (algebraization of an isolated analytic germ). Let
(X, x) be a normal complex analytic germ of dimension d >= 1,
smooth away from x, with free holomorphic tangent module of rank d.
There exist a finitely generated normal integral C-algebra A and a
maximal ideal m in A such that
dim A = d, A/m = C,
A-hat_m = O-hat_{X, x},
Der_C(A) = A^d.
The affine variety Spec A can be chosen smooth away from m. If x is
singular, A_m is nonregular.
Proof. Completed analytic presentation. Embed the germ in a smooth
analytic germ and write
S = O_{X, x} = C{x_1, ..., x_e} / I_an.
Choose a finite row f generating I_an and a finite matrix D that
generates all its relations over the convergent power-series ring.
Completion is exact on finite modules, so these give the
presentation of the finite-determinacy lemma over
P = C[[x_1, ..., x_e]], with R = S-hat = P/(f). Complex analytic
local rings are excellent (Cutkosky, "Etoiles and valuations",
J. Pure Appl. Algebra 221, 2017, 588-610). Hence the completion of
the normal local ring S is a normal local domain, and
dim R = dim S = d, R regular if and only if S regular.
Convergent conormal dual. The convergent conormal dual of the
analytic embedding coincides with Hom_R(I/I^2, R). On the smooth
locus the tangent-to-normal map is surjective, since a smooth
analytic subspace of a complex manifold is locally a regular
embedding. Its cokernel is a coherent module supported at the
isolated point x, and has finite length. Flat completion identifies
this cokernel with T^1 = K/J above. Finite determinacy therefore
applies to f, D.
Continuous tangents. The holomorphic differentials of S have the
finite Jacobian presentation with generators dx_j and relations
df_i; their dual is the holomorphic tangent module. Flat completion
of this kernel gives, where continuous means with respect to the
maximal-ideal topology,
Der_C^cont(R, R)
= ker( R^e -> R^s, (v_j) |-> sum_j v_j d_j f-bar )
= R^d.
A continuous derivation of a power-series quotient is determined by
its values on the x_j, and the displayed relations are precisely
the conditions for it to descend from P.
Approximation of equations and relations. Let
H = ( C[x_1, ..., x_e]_{(x_1, ..., x_e)} )^h
be the henselization, whose completion is P. This is the
henselization of a finite-type C-algebra at a prime, as covered by
Artin's theorem. Apply that theorem over H to the polynomial system
in the s + s t entries of a row G and a matrix B,
sum_{i=1}^s G_i B_{i nu} = 0 (1 <= nu <= t),
with formal solution (G, B) = (f, D). It gives g in H^s and B in
H^{s x t} with gB = 0 exactly and
g - f in n^{n_0} P^s, B - D in n^N P^{s x t}.
By the finite-determinacy lemma, P/(g) = R.
The henselization is the filtered colimit of pointed etale
neighborhoods of the origin, with the same residue field. The
finitely many entries of g, B are represented on a single such
neighborhood. Passing to a later neighborhood, the identities
gB = 0 hold there as well. Finite presentation and clearing
denominators spread this to a neighborhood of the origin in affine
space; shrinking preserves the marked point and makes the morphism
etale. After an affine shrinking the identities hold in the
coordinate ring. This gives an affine etale neighborhood
Spec E -> A^e_C with a marked point q over the origin, residue
field C, and a row g of E vanishing at q. Its completed ambient
local ring is P. Put
A' = E/(g), p' = q/(g), T = A'_{p'}.
Then A' is of finite type over C, and exactness of completion
together with P/(g) = R gives
T-hat = P/(g) = R.
So this construction preserves the complete local ring, including
its dimension and its normality.
Descent of tangent freeness. Because E is etale over A^e_C, its
differentials have basis dx_j near q. Completing the finite
Jacobian presentation of Omega^1_{T/C} therefore gives the
continuous Jacobian presentation of P/(g). For a finitely presented
module, flat base change commutes with its dual (express the dual
as the kernel of a map between finite free modules). This gives
Der_C(T) tensor_T T-hat = Der_C^cont(T-hat, T-hat) = T-hat^d.
Let M = Der_C(T). The residue vector space of M has dimension d.
Lift a basis to d elements of M. Nakayama's lemma gives a
surjection T^d -> M; after completion this is a surjection between
free modules of rank d over T-hat, hence an isomorphism. Faithful
flatness of completion annihilates its kernel, proving M = T^d.
Local ring properties. The ring T injects into its completion,
which is a domain, so T is a domain. Normality also descends: if
a/b in Frac(T) is integral over T, it is integral over the normal
T-hat, hence lies in T-hat; then a in b T-hat intersect T = b T,
the last equality by faithful flatness, so a/b in T. Completion
preserves Krull dimension and the dimension of m_T/m_T^2. So T has
dimension d and is nonregular whenever S is.
Isolated singularity. Let j_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 m_S lies in j_an. Fitting ideals commute with base change.
Completed Jacobian presentations plus T-hat = R therefore imply,
by contraction from the faithfully flat completion,
m_T^b is contained in Fitt_d(Omega^1_{T/C})
for some b >= 0. On the reduced pure-d-dimensional neighborhood
constructed next, the algebraic Jacobian criterion interprets this
as nonsmoothness supported only at the marked point.
Integral normal affine neighborhood. The finite module Der_C(A')
localizes at p' to Der_C(T), because Omega^1_{A'/C} is finitely
presented. Represent the chosen basis after inverting an element
outside p'; this defines a map from a free module of rank d to
Der_C(A') whose kernel and cokernel vanish at p'. Their supports
are closed, so a further principal shrinking makes this map an
isomorphism.
Since T is a domain, exactly one irreducible component of
Spec A' passes through p', and the support of the nilradical
avoids p'. Remove the other components and that support. The
remaining neighborhood is reduced and irreducible. Its normal locus
is open (finite type over C is excellent) and contains p' by the
normality of T; restrict there. This neighborhood has dimension d,
and its nonsmooth locus is closed; by isolated descent it has no
positive-dimensional component through p'; remove the components
that do not contain p'.
All restrictions are open neighborhoods of p'. Choose a principal
affine neighborhood contained in their intersection and write it
Spec A, with marked maximal ideal m. Then A is finitely generated,
normal, and integral, dim A = d, and
A_m = T, Der_C(A) = A^d.
The residue field and completed local ring are unchanged, and the
variety is smooth away from m. []
Main Theorem. There exist a finitely generated normal integral
C-algebra A of Krull dimension two and a maximal ideal m in A such
that
Der_C(A) = A^2 and A_m is not regular.
Proof. The analytic counterexample supplies a singular normal
complex analytic surface germ, smooth away from its marked point,
whose holomorphic tangent module is free of rank two. Apply the
algebraization proposition and fix any of the approximations and
affine neighborhoods furnished by its proof. Its coordinate ring A
is a finitely generated normal integral C-algebra of dimension two
with Der_C(A) = A^2 and A_m nonregular. The derivation module is
free, in particular projective, while A is not regular. []
The smooth locus. Let X = Spec A be the affine surface just
constructed, and U = X_reg its smooth locus. The logarithmic
Kodaira dimension of a smooth quasiprojective surface U is
kappa-bar(U) = kappa(K_Ubar + D),
where Ubar is a smooth projective compactification and its reduced
boundary D = Ubar minus U is a simple normal-crossing divisor. The
value does not depend on the compactification.
Corollary. kappa-bar(U) = 2.
Proof. The free tangent sheaf on X restricts to a trivial tangent
bundle on U. If kappa-bar(U) <= 1, the theorem of Biswas, Gurjar,
and Kolte (J. London Math. Soc. 90, 2014, 270-286) would make the
normal affine surface X smooth, contradicting the Main Theorem.
The only remaining value of logarithmic Kodaira dimension for a
surface is two. []
Derivation blowups. For a finite A-module M of generic rank r, its
blowup 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. The universal property is described in
Barajas, Chavez-Martinez, and Romano-Velazquez, "The blow-up of a
singularity at the module of derivations" (2026;
arXiv:2510.07607v2), section 1.6.
Corollary (derivation blowups). Starting with the singular
surface X of the Main Theorem, 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.
Proof. The derivation module is free, so the identity morphism of
X already has the universal property stated above. Its blowup at
the derivation module is therefore X itself, and the operation
remains the identity at every iteration. Normalization also leaves
X unchanged, because A is normal. []
The surface X is Gorenstein, its canonical divisor is zero, and
the canonical discrepancy at the exceptional curve is -5. In the
resolution, the exceptional curve has genus g = 21, self-
intersection -10, and first Betti number b = 0 in the dual graph.
The geometric genus is at least 33. Hence p_g - g - b >= 12,
which exceeds the bound p_g - g - b <= 2 of Bergner and Graf
(Forum Math. Sigma 8, 2020, paper e21). The surface is not log
canonical.
The surface X stands outside every positive result on the
Lipman-Zariski conjecture for normal surfaces. Its tangent sheaf
is free, but no frame can be made commuting at the singular point.
This follows from the theorem of Jorder (Math. Z. 278, 2014,
893-899) stating that a normal complex space with a local frame of
pairwise commuting vector fields is smooth at the point. The
surface is not covered by the log canonical extension theorem of
Greb, Kebekus, Kovacs, and Peternell, by Druel, or by Graf and
Kovacs, because its canonical discrepancy is -5, and it is not
covered by the geometric-genus criteria of Graf or of Bergner and
Graf, because p_g - g - b >= 12.
Derivation blowups at the tangent module, and derivation blowups
followed by normalization, leave X unchanged and do not resolve
its singular point, so the Lipman-Zariski conjecture fails in
characteristic zero for normal surfaces, and the analogous question
of Barajas, Chavez-Martinez, and Romano-Velazquez about derivation
blowups has a negative answer in characteristic zero.