editors: nobody
PIXTON COMPLETENESS: A COUNTEREXAMPLE
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PART 0: WHY YOU MIGHT CARE
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Here is the whole paper in one sentence:
Mathematicians guessed a complete set of rules for measuring
curves. We found one measurement their rules cannot explain --
and yet that measurement still comes out equal to zero when we
actually compute it.
The rest of Part 0 unpacks that sentence.
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0.1 What is a curve, in math?
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A curve, in the algebraic-geometric sense used in this paper, is a
shape that locally looks like the real line (a real curve) or the
complex plane (a complex curve). Key examples:
- a circle;
- a sphere, which is a real surface (dimension 2 over R);
- a torus, which is the surface of a donut;
- an n-holed donut, for any n >= 0.
"Genus" is the number of holes. A sphere has genus 0, a torus has
genus 1, a two-holed donut has genus 2, and so on.
A complex curve is a surface where we additionally know how to add
and multiply points (in a way compatible with the shape). This paper
works with smooth compact complex curves of a fixed genus g, plus a
finite set of "marked points" (distinguished labeled locations on
the curve, numbered 1 through n).
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0.2 What is "the space of all curves of genus g"?
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Take all curves of a fixed shape -- say, all genus-g curves with n
marked points. This is not one curve; it is a family of curves that
itself varies. In fact this family has the structure of a shape (a
"space") in its own right: it is called the moduli space and is
written M_{g,n}.
A point in M_{g,n} is "one curve-with-markings." The whole space
M_{g,n} is all curves packed together, roughly like a rainbow of
every possible genus-g shape smoothly varying between its neighbors.
M_{g,n} is not compact -- you can let a curve "degenerate" by
pinching a bit of it down to a point. That produces a boundary;
adding the boundary produces the compact moduli space of stable
curves, Mbar_{g,n}. Stable curves are the curves that appear at
the boundary: ordinary smooth curves plus curves with a limited
kind of singularity called a node.
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0.3 What can we measure on a curve?
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At every marked point i there is a natural line bundle of tangent
directions, whose first Chern class is written psi_i. There are
also kappa classes kappa_j that measure how the marked points fall
around a curve.
The important point: psi and kappa are canonical, universal
measurements. Every smooth curve has them; they vary continuously
as the curve varies through moduli.
Beyond psi and kappa, we can build more measurements by multiplying
psi's and kappa's, and by pushing forward along forgetful maps
(forget a marked point) and gluing maps (glue two marked points,
creating a node). All measurements built this way are called
tautological classes.
The set of all tautological classes is a ring called the
tautological ring, written R^*(Mbar_{g,n}).
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0.4 What is a "relation"?
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A relation in the tautological ring is a rule that says: "whenever
you compute these particular measurements combinatorially, they add
up to zero."
Equivalently, a relation is a formal tautological class that
actually evaluates to zero when you test it against every cycle.
Two things must be distinguished carefully:
1. A formal combination of measurements. This lives in the
strata algebra S_{g,n} -- a giant free abelian group on formal
symbols. It has no geometry inside it; it is pure bookkeeping.
2. The realization map q: S_{g,n} -> A^*(Mbar_{g,n}),
which substitutes the actual psi and kappa classes for their
formal symbols.
A proposal for "a complete list of relations" is a proposal for
what q's kernel is: the set of formal combinations that always
evaluate to zero.
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0.5 What did Pixton propose?
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In 2012 Aaron Pixton proposed a large explicit family of formal
relations. His relations look like sums over stable graphs,
decorated with specific generating series. He guessed that the only
tautological relations were the ones his rules produce, possibly
multiplied and moved around by formal gluing operations.
Concretely: let P_{g,n} be the formal span of all Pixton relations
plus all their formal consequences. Pixton's conjecture (for the
original system) is:
ker q = P_{g,n}
for every stable pair (g,n).
In words: every formal combination that vanishes in actual Chow
theory should be expressible using Pixton's rules plus gluing.
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0.6 What this paper proves
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This paper constructs a stable pair (g,n) with g = 10^60 and
n = 3 * C(g,3), and a specific formal class Y in S_{g,n}, with
two properties:
(a) q(Y) = 0 in actual Chow theory -- so Y is a real relation,
not a formal wish;
(b) Y is NOT in Pixton's span P_{g,n}.
Together, (a) and (b) say:
ker q_{g,n} is STRICTLY BIGGER than P_{g,n}.
Pixton's conjecture, in its "original system" form, is false.
Because cycle class commutes with everything in sight, part (b)
also shows the analogous statement in rational cohomology: Y is
a cohomological relation that Pixton's list does not generate.
That is what the paper is about. Parts 1 through 6 build the
machinery to prove (a) and (b). Part 7 discusses the context.
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PART 1: BACKGROUND
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This part puts the standard definitions on the table. Skip
ahead to Part 2 if you already know this material. Read on
if you want plain-English context for everything that follows.
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1.1 Curves, markings, and stability
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Fix a base field, here C (complex numbers), and a genus g >= 0.
A smooth curve of genus g over C is a compact smooth complex
curve of genus g. Equivalently, it is a compact Riemann
surface of genus g.
A pointed curve (C; p_1,...,p_n) has n labeled, distinct
points p_1,...,p_n on C.
Not every (g,n) can have a marked curve that degenerates
nicely: we need (g,n) to be stable, meaning
2g - 2 + n > 0.
Stable pairs with g >= 2 can have any n >= 0. The pair (0,2)
is not stable, and (0,0), (0,1) are not stable either.
A stable curve of genus g with n marked points is a connected
curve C that is allowed to have nodes (transverse self-
intersections; two branches meeting at one point) subject to:
(a) every smooth component is smooth;
(b) marked points are smooth points and distinct;
(c) every rational component (genus 0) has at least three
"special" points (nodes plus markings).
Condition (c) is exactly the "stability" condition on each
component.
The moduli space Mbar_{g,n} is a smooth proper Deligne-Mumford
stack parametrizing stable curves of genus g with n marked
points. Its coarse space is a projective variety of dimension
3g - 3 + n. The open subspace M_{g,n} of smooth curves is a
Zariski-dense open.
We work over C throughout. We work with rational Chow groups
and rational cohomology.
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1.2 Stable graphs
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Every stable curve has a "combinatorial shadow" called its
stable graph. A stable graph Gamma of type (g,n) consists of:
- a finite set V(Gamma) of vertices;
- a finite set E(Gamma) of edges (possibly loops); a loop
is an edge with both half-edges at the same vertex;
- a finite set of half-edges, with each edge having two
half-edges and with a "leg" attached to exactly one
vertex for each labeled mark;
- a genus g_v >= 0 at each vertex v;
- a total genus condition sum_v g_v + |E| - |V| + 1 = g;
- a labeling of the n "legs" by the marks 1,...,n;
- per-vertex stability 2g_v - 2 + n_v > 0, where n_v is
the number of half-edges incident to v.
Every vertex of a stable graph corresponds to an irreducible
component of the corresponding stable curve; every edge to
a node.
The gluing map xi_Gamma: product_v Mbar_{g_v,n_v} -> Mbar_{g,n}
glues the appropriate cotangent lines together at each node.
Its image is the closed subset of Mbar_{g,n} consisting of
curves with dual graph refining Gamma.
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1.3 What is a Chow group (very briefly)
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A smooth projective variety X over C has a Chow group A^*(X),
an algebraic analog of cohomology. Roughly, A^k(X) is
"codimension-k algebraic cycles modulo rational equivalence."
Two facts matter here:
- A^*(X) is a graded commutative ring: one can intersect
cycles and add them.
- For X = Mbar_{g,n}, cycle class maps A^k(X) -> H^{2k}(X)
compatibly identify many classes with cohomology.
This paper lives almost entirely in A^*(Mbar_{g,n}). Chow
groups are the natural setting for the whole story because
tautological pushforwards, pullbacks, and intersections all
live there. The cycle-class map carries everything over to
rational cohomology without loss of the counterexample.
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1.4 The strata algebra S_{g,n}
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We now define the formal object in which "the Pixton span"
lives.
Take a decorated stable graph Gamma of type (g,n). To Gamma
attach a decoration
gamma = product over half-edges of psi_h^{a_h}
* product over vertices v of kappa_{j}^{c_{v,j}},
with finitely many nonzero c_{v,j}. At each vertex, impose
deg(gamma|_v) <= 3 g_v - 3 + n_v.
This "per-vertex dimension bound" is the standard truncation
used by Pandharipande-Pixton-Zvonkine. It's important: the
strata algebra is graded, and we always truncate to degree no
higher than the local dimension at each vertex.
The strata algebra S_{g,n} is the free Q-vector space on
isomorphism classes of decorated graphs [Gamma, gamma], modulo
graph isomorphism, with the truncation above. It is graded by
total degree = |E(Gamma)| + deg(gamma).
There is a linear map
q_{g,n}: S_{g,n} -> A^*(Mbar_{g,n}; Q),
[Gamma, gamma] |-> xi_{Gamma,*} gamma,
the gluing pushforward. Its image is the tautological ring:
R^*(Mbar_{g,n}) = q_{g,n}(S_{g,n}).
This gluing pushforward is unnormalized: no automorphism
denominators, exactly the convention of PPZ (Pandharipande-
Pixton-Zvonkine). Whenever automorphism denominators occur
in this paper, they are explicit in the class being defined;
they do not modify q_{g,n}.
The realization q_{g,n} is Q-linear and degree-preserving. It
is not injective: many nonzero formal combinations map to zero
in A^*(Mbar_{g,n}). Its kernel is exactly the collection of
all tautological relations.
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1.5 Products in the strata algebra
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There is a natural product on S_{g,n}, compatible with q. It
is worth remembering how it works, because several proof steps
compose formal operations.
Given two decorated graphs, one first takes a "common
refinement" -- a graph that maps compatibly to both by edge
contractions, with the source symmetries of the contractions
fixed. Then:
- edges that survive in at least one contraction keep their
psi decorations;
- edges that get "double-counted" (both contractions want to
keep the edge but the intersection shortens it) pick up a
normal factor of -psi - psi', i.e. the negative sum of
the cotangent classes at the two half-edges;
- leg and flag decorations pull back to their components;
- kappa classes split additively (e.g. kappa_j at a vertex
pulled to a split becomes sum over the two halves);
- if any vertex exceeds its dimension, the whole term
becomes zero.
This product is symmetric and associative. It is the "transverse
and self-intersection" rule for node cuts, exactly the formal
version of the standard boundary calculus.
Because we always truncate at the local vertex dimension, the
product lands in S_{g,n}. This is one place the truncation
matters: excessive decorations remain excessive after splitting.
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1.6 Forgetful and gluing operations
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Two families of extra structural maps make S_{g,n} into a
"tautological" system in the technical sense:
- Forgetful pullback pi_y^*: S_{g,n} -> S_{g,n+1}, adding
a new leg y at every vertex in all possible ways;
- Forgetful pushforward pi_{y,*}: S_{g,n+1} -> S_{g,n},
forgetting the leg;
- Gluing pushforward gl_*: S_{g,n+2} -> S_{g,n} gluing two
legs to create a node;
- Gluing pullback gl^*: S_{g,n} -> S_{g,n+2} (with a bit of
care) going the other way.
All these are q-compatible: q_{g,n}(pi_y^* s) = pi_y^*(q_{g,n+1}(s))
and similarly for the rest. This is the "tautological" pillar
of the strata algebra. It means any operation we can perform in
Chow can also be performed formally.
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1.7 Pixton's relations, precisely
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Now the star of Part 0 returns in formal dress.
Set
A(T) = sum_{m >= 0} (6m)! / ((3m)! (2m)!) T^m
B(T) = sum_{m >= 0} ((6m+1)/(6m-1)) * (6m)! /
((3m)! (2m)!) T^m
The coefficients c_m = (6m)!/((3m)! (2m)!) appear throughout the
tautological literature: they are the coefficient weights of
Faber-Zagier.
For a commuting parity variable zeta with zeta^2 = 1, set:
Chat_{3j}(T, zeta) = T^j A(zeta T)
Chat_{3j+1}(T, zeta) = zeta T^j B(zeta T)
Indices congruent to 2 mod 3 are never used.
Pixton's relation system, at a decorated graph Gamma of type
(g,n), is built out of these. Give each vertex its own parity
variable zeta_v. The recipe:
- To every kappa factor K_{e, epsilon} assign a generating
series coefficient via the "cycle operator"
kappaHat_Gamma( product_i K_{e_i, epsilon_i} )
= sum_{(cycles c of tau)} product over c of
(sum over v of kappa_{sum_{i in c} e_i}^{(v)}
* zeta_v^{sum_{i in c} epsilon_i})
summed over all permutations tau in S_l. This is the
standard Faber formula for kappa monomials.
- To every edge e = (v, w) with cotangent symbols x, y at
the two half-edges, assign the "edge generating series"
Delta_e = [ A(zeta_v x T) * zeta_w B(zeta_w y T)
+ zeta_v B(zeta_v x T) * A(zeta_w y T)
+ zeta_v + zeta_w ] / ((x + y) T).
This is a power series in T by the identity
A(T) B(-T) + A(-T) B(T) + 2 = 0.
- To every mark i at vertex v(i) with cotangent psi_i, assign
Chat_{a_i}(psi_i T, zeta_{v(i)})
for a nonnegative integer a_i (the "mark index").
- To a partition sigma = (sigma_1, ..., sigma_l), each part
not congruent to 2 mod 3, assign a product of
Chat_{sigma_j} contributions at the vertex series.
Now choose positive integers r (degree), a partition sigma with
no parts congruent to 2 mod 3, and marks a_1,...,a_n each not
congruent to 2 mod 3, subject to the "admissibility" conditions
3r >= g + 1 + |sigma| + sum_i a_i
3r = g + 1 + |sigma| + sum_i a_i mod 2.
Then the Pixton relation at (g,n,r; sigma, a) is the formal sum
R(g,n,r;sigma, a)
= sum over Gamma [Gamma, P_Gamma] / |Aut Gamma|
where P_Gamma is the coefficient of
T^{r - |E(Gamma)|} * product_v zeta_v^{g_v + 1}
in the graph series:
2^{-h^1(Gamma)} * kappaHat_Gamma(
exp{1 - Chat_0} * product_{j=1}^l {Chat_{sigma_j}} )
* product_i Chat_{a_i}(psi_i T, zeta_{v(i)})
* product_{e in E} Delta_e .
Here h^1(Gamma) = |E| - |V| + 1 is the first Betti number of
the graph (its loop count), and "{series}" means: expand in
T and zeta, take the specified coefficient, and substitute
into K_{m, epsilon} kappa-monomial coefficients.
This is the recipe. If it looks baroque, that's because it is;
the point is not to memorize it but to see that it is a single,
explicit, uniform recipe for building formal combinations of
stable graphs out of three ingredients:
(a) generating-series coefficients from A(T), B(T);
(b) kappa operators;
(c) edge functions.
Pixton's crucial observation is that the resulting recipe
turns out to produce genuinely vanishing classes when realized
through q.
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1.8 The Pixton span P_{g,n}
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Pixton's relation recipe gives many formal combinations. To
turn them into a machinery capable of representing "all
formal consequences of Pixton's rules," we build the Pixton
span by:
1. Taking a Pixton relation at some vertex v_0 of an outer
stable graph Gamma of type (g,n), placing arbitrary
decorated strata classes at the other vertices, and
gluing via the product rule.
2. Taking the span of all such combinations, over every
choice of outer graph, vertex, and arbitrary auxiliary
decorations.
This P_{g,n} is already an ideal: Pixton's Proposition 1
shows it is stable under the forgetful and gluing operations
of Section 1.6. This is exactly the sense in which P is
the natural "span of consequences" rather than just a
specific family of relations.
The smaller "0/1 marking" seed system, where all a_i in {0,1}
and sigma is empty, is a subsystem that PPZ (Section 3.5)
shows builds the entire span P_{g,n} from these seeds by
multiplication, forgetting, and gluing.
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1.9 What Pixton actually conjectured
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Pixton asked: is P_{g,n} the full kernel of q_{g,n}?
Equivalently: does
P_{g,n} = ker q_{g,n}
hold for every stable (g,n)?
The known direction is
P_{g,n} is a subset of ker q_{g,n}.
This is the validity theorem, which has been known for a
while now: PPZ proved it for rational cohomology via the
3-spin CohFT, Janda proved it in Chow via equivariant
Gromov-Witten theory of P^1. The counterexample in this
paper refutes the reverse inclusion: P is strictly smaller
than ker q for at least one (g,n).
So the setup is:
P_{g,n} subseteq ker q_{g,n} subseteq S_{g,n},
^
this paper shows strict inclusion
at g = 10^60, n = 3*C(g,3).
Two points deserve emphasis in this rewrite.
First: the conjecture is about ALL (g,n) at once, not
a single pair. To refute it, one stable pair suffices.
That's exactly what Part 3 constructs.
Second: P_{g,n} is a fixed, explicit object. Failure
of the conjecture is not vague: it means there is an
explicit class Y in the fixed S_{g,n} with q(Y) = 0
and Y not in P_{g,n}. That's a concrete, checkable
statement.
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1.10 Two flavors: Chow and rational cohomology
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The paper proves the failure in two ways:
- Chow: q_{g,n}(Y) = 0 in A^*(Mbar_{g,n}), still Y not
in P_{g,n}. This is Theorem 2.1 (stated in Part 2 below).
- Rational cohomology: q^H_{g,n}(Y) = 0 in
H^*(Mbar_{g,n}(C); Q), still Y not in P_{g,n}. This is
the corollary.
The second follows from the first: apply the rational cycle
class map cl: A^*(Mbar_{g,n}) -> H^*(Mbar_{g,n}(C); Q).
Compatibility of cl with Chern classes, products, and proper
pushforwards shows cl(q_{g,n}(s)) = q^H_{g,n}(s) for all s.
So whenever q_{g,n}(Y) = 0, automatically q^H_{g,n}(Y) = 0.
Nonmembership Y not in P_{g,n} is a single assertion in S_{g,n},
unchanged whether we then pass to A or to H.
End of Part 1. Part 2 states the main theorem in full.
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PART 2: THE MAIN THEOREM
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Now the formal statement. Everything in Parts 3 through 6
builds up the pieces of Theorem 2.1 and Corollary 2.2 below.
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2.1 The main theorem
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Theorem 2.1. Set
g = 10^60, D_* = C(g,3), n = 3 D_*.
Here C(g,3) is the binomial coefficient "choose 3 from g". So
D_* is the number of increasing triples from {1,...,g}, and
n = 3 D_* is exactly three times that (because each triple
gets three labeled slots).
There is an explicit class
Y in S_{g,n}^{2 D_*}
of Definition 3.4 (constructed in Part 3) satisfying
q_{g,n}(Y) = 0 in A^{2 D_*}(Mbar_{g,n}; Q),
Y not in P_{g,n}.
Consequently Pixton's original relation span does not exhaust the
tautological Chow relations: the equality ker q_{g,n} = P_{g,n}
fails for the stable pair (g, n) above.
The class Y has formal degree 2 D_*: it is a product of D_*
"factors" each of codimension two. This is far below the top
dimension 3g - 3 + n of S_{g,n}, so Y is a codimension-low
class, not a top-degree class.
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2.2 The cohomological corollary
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The same class gives the rational-cohomological statement.
Define, in formal degree d,
q^H_{g,n}: S^d_{g,n} -> H^{2d}(Mbar_{g,n}(C); Q)
q^H_{g,n} = cl o q_{g,n},
where cl is the rational cycle class map for the complex Deligne-
Mumford stack Mbar_{g,n}. Its image is the cohomological
tautological ring RH^* = cl(R^*) inside H^*(Mbar_{g,n}(C); Q).
Corollary 2.2. For the stable pair and class Y of Theorem 2.1,
q^H_{g,n}(Y) = 0 in H^{4 D_*}(Mbar_{g,n}(C); Q),
Y not in P_{g,n}.
In particular, P_{g,n} is a proper subset of ker q^H_{g,n}.
Proof of Corollary 2.2.
Cycle class commutes with Chern classes, with products, and with
proper pushforward. In particular
cl(xi_{Gamma,*} gamma) = xi_{Gamma,*} cl(gamma)
for any decorated graph (Gamma, gamma), and the same compatibility
applies to the forgetting-pushforward definition of kappa classes.
Therefore q^H_{g,n} is exactly the cohomological graph realization
of Pandharipande-Pixton-Zvonkine's Section 0.3, with the same
vertexwise truncation and unnormalized generators. Applying cycle
class to the equality q_{g,n}(Y) = 0 from Theorem 2.1 gives
q^H_{g,n}(Y) = 0.
Nonmembership Y not in P_{g,n} is an assertion in the formal
strata algebra S_{g,n}; it is untouched by passing from A^* to
H^*. The strict inclusion
P_{g,n} subseteq ker q^CH_{g,n} subseteq ker q^H_{g,n}
then gives the claimed statement. Passing from Chow vanishing to
cohomological vanishing uses the second inclusion; it requires no
injectivity of cycle class.
End of Part 2. Part 3 builds the class Y.
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PART 3: THE CLASS Y
=====================================================
We now construct the class Y whose existence Theorem 2.1
promises. The construction is purely combinatorial:
we assemble Y out of D_* = C(g,3) "factors", each of which
is a small correction to a background class.
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3.1 The global parameters
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Recall from Theorem 2.1:
g = 10^60, D_* = C(g,3), n = 3 D_*.
The number g is astronomically large. This is not important
mathematically except that g has plenty of room compared to
certain other constants that appear in the proof. We say more
about this in Part 5.
The number D_* counts increasing triples of distinct colors
1 <= i < j < k <= g.
Each such triple will contribute exactly one factor to Y.
Each factor has codimension two, so Y has codimension 2 D_*.
The number n equals exactly 3 D_*: three labels per increasing
triple. So the parameters line up: one factor per triple, three
slots per factor, n total slots.
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3.2 The color word
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Order the increasing triples of colors lexicographically. For
example, if g >= 4, the very first triples in lexicographic
order are
(1,2,3), (1,2,4), (1,2,5), ..., (1,3,4), (1,3,5), ...
Write the colors in each successive triple down in that order
to form one long word
i = (i_1, i_2, ..., i_n)
of total length n. The first three entries are 1,2,3 (or, if
g < 3, we wouldn't be working in this regime). The next three
are 1,2,4. And so on. Each color j from 1 to g appears in the
word exactly
C(g-1, 2)
times: choose two other distinct colors to complete the triple.
Group the labels 1,...,n into consecutive triples J_1, ..., J_{D_*},
each of size three, aligned with the word i above. So the J-th
block J = (a, b, c) is exactly a triple of slots whose color
word at those slots is some increasing triple (i, j, k) of colors.
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3.3 Tail divisors D_T
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For a subset T of the labels with |T| >= 2, let D_T denote the
divisor class in A^1(Mbar_{g,n}) of stable curves where the marks
in T cluster together on a rational tail and are separated from
the rest of the curve by a single node.
Informally: a genus-zero sphere sprouts out, holds exactly the
labels T, and is attached to the main curve by a single node.
For distinct labels a, b, define
D_bar_{ab} = sum over T that contains {a, b} of D_T.
So D_bar_{ab} is the sum of all rational tail divisors containing
both a and b. Rational tails containing only a, or only b, do
not appear.
Both D_T and D_bar_{ab} are codimension-one classes. We will
multiply them by other classes; each factor F_J^st below
involves exactly two such sums multiplied together, giving
codimension two per factor.
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3.4 Rational bridges
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Beyond rational tails, we need a second kind of boundary class:
a rational bridge. A rational bridge is a rational component
that separates two parts of the curve: a sphere with two
attached nodes on opposite sides.
Concretely, a "one-edge unmarked stable graph of genus g" is a
stable graph with exactly one edge (a node), two vertices
whose genera sum to g, and no marks. Subdividing that edge
by inserting a fresh genus-zero vertex between them and placing
all three labels of a block J at the new vertex produces a
graph we can call a three-mark rational bridge. Summing over
all such choices, dividing by the automorphism order of each
graph, and pulling back to forget everything outside J gives
the formal class
beta_J^st in S_{g,n}^2.
This is the bridge correction.
One clarifying point: the sum beta_J^st involved is over
unordered pairs of vertices. The 1/|Aut Gamma| normalizations
in its definition are the graph-automorphism weights, not
something imposed on the unnormalized generators of S_{g,n}.
They are exactly what makes beta_J^st a natural class with
respect to the forgetful pushforward, but they do NOT alter
the formal product rule (which stays unnormalized).
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3.5 The three-slot factor F_J^st
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For a block J = (a, b, c) with three slots, define
F_J^st = D_bar_{ab} * D_bar_{ac} - (1/2) * beta_J^st.
This is the building block of Y. It has two terms:
- a product of two codimension-one tail divisor sums;
- minus one half times a codimension-two bridge correction.
The coefficient -1/2 is essential: it is what makes the
eventual compactification of F_J^st match a projected small
diagonal over the boundary (see Part 5). Without the
correction, a codimension-two exceptional class would fail to
cancel. The coefficient is not adjustable.
Notice the asymmetry in the first term: D_bar_{ab} * D_bar_{ac}.
Both sums contain a. The reason for this choice: we want a
class sensitive to "a agrees with b AND a agrees with c", which
is the smallest interesting arrangement of three labels.
F_J^st has total formal degree 2 (one from each D_bar term, or
two from beta_J^st). It lives in S_{g,n}^2.
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3.6 The alternation and the definition of Y
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The class F_J^st is not symmetric under arbitrary permutations
of the three labels a, b, c of J. It singles out a (via the
{ab, ac} pattern).
The remedy is alternation. Let
G_i = { tau in S_n : tau(i) = i }
be the set of permutations of labels that preserve the color
word i. Equivalently, tau in G_i shuffles labels within
color-classes: colors are fixed but slots holding the same
color can be permuted among themselves.
Definition 3.4. The candidate class Y is
Y = sum over tau in G_i of
sgn(tau) * tau( product_{J} F_J^st ).
Here tau acts on labels: it permutes the slots of each factor
F_J^st and permutes which factor goes where, but it preserves
the color word i because tau lies in G_i. The sign sgn(tau) is
the ordinary permutation sign.
Because G_i preserves the word, the colors of the slots are
unchanged. The sum thus alternates over S_n-internal shuffling
without disturbing the color pattern.
Informally: Y is the "signed-average over relabelings of colors"
of product_J F_J^st. The alternation makes Y into a genuine
antisymmetric tensor in the colors, not just a naive product.
This is Definition 3.4 (equivalently Definition 1.2 of the
original paper). It is the class Theorem 2.1 talks about.
--------------------------------------------------------------------------------
3.7 Sanity checks
--------------------------------------------------------------------------------
Before doing anything with Y, we check that it is well defined
and lives where we think it does.
(a) Finiteness. Every operation in Definition 3.4 is a finite
combination: the set G_i is finite (it is a subgroup of S_n,
whose order is huge but finite), and each F_J^st has
finitely many terms. So Y is a genuine element of S_{g,n}.
(b) Degree. Each factor F_J^st has codimension two. There are
D_* = C(g,3) factors. Straight multiplication gives
deg(Y) = 2 D_*. This is far less than 3g - 3 + n, the top
dimension of S_{g,n}; so Y is a strictly positive-codimension
class.
(c) Stability of (g,n). Since g = 10^60 >= 2 and n = 3 D_*
is positive, the pair (g,n) is stable.
(d) Automorphism weights in beta_J^st. The 1/|Aut Gamma| in
beta_J^st specifies the coefficients of a particular class
in S_{g,n}. It does NOT modify the unnormalized generators
q_{g,n}[Gamma, gamma] = xi_{Gamma,*} gamma. Both facts are
compatible: the product rule for S is unnormalized, and
arbitrary rational coefficients are allowed on top of it.
None of these checks is deep. They are here to reassure the
reader that Y is a legitimate element rather than a formal
wish.
--------------------------------------------------------------------------------
3.8 Roadmap: how Theorem 2.1 splits
--------------------------------------------------------------------------------
Proving Theorem 2.1 splits into two completely independent
claims, proved by completely different methods:
Claim A (non-membership): Y is not in P_{g,n}.
Claim B (vanishing): q_{g,n}(Y) = 0.
A and B involve different algebras and different techniques:
- Claim A uses a "formal detector": a Q-linear map out of
S_{g,n} that annihilates P_{g,n} but takes nonzero values
on Y. The detector can be constructed entirely within the
strata algebra; it doesn't touch actual Chow geometry.
This is Part 4.
- Claim B uses a sophisticated analysis of relative Chow
correspondences on curve powers, surface Quot schemes,
a combinatorial identity from Hilbert schemes of points,
and a support-dimension induction over the boundary.
This is Part 5 (surface relation) plus Part 6
(compactification and support).
These two claims are independent. Part 4 only uses the formal
product rule; Part 5-6 only use actual Chow geometry. It is our
claim that both facts are true, and together they prove
Theorem 2.1.
End of Part 3. Part 4 builds the formal detector.
=====================================================
PART 4: WHY Y IS NOT IN THE PIXTON SPAN
=====================================================
Part 4 constructs Claim A of Section 3.8: a linear map that
vanishes on the Pixton span but does not vanish on Y. That
proves Y is not in P_{g,n}.
Our tool is an idea that is entirely algebraic. We build a
special algebra -- a "Frobenius algebra" -- that is only allowed
to see a particular kind of stable graph (a "rational tree").
On rational trees, this Frobenius algebra gives us a clean
recipe for testing formal combinations. Any Pixton relation,
restricted to rational trees, evaluates to zero under this
recipe. But Y, restricted to rational trees, evaluates to a
specific nonzero element.
The trick is to prove this in one decisive step: an exterior
algebra R with one "color" generator per increasing triple of
curve-colors. It is exactly the kind of algebra where a
distinguished product over all triples -- like Y evaluated
color-by-color -- is nonzero.
A first reading: skim 4.1-4.3 for the setup, take on faith that
4.6 defines a linear map, and look at Part 4.8 (which is where
the punch line is). Come back to 4.4-4.7 later if you want the
manipulative details.
--------------------------------------------------------------------------------
4.1 The exterior coefficient ring
--------------------------------------------------------------------------------
Let
R = Lambda_Q ( theta_{ijk} : 1 <= i < j < k <= g )
be the exterior algebra on one generator theta_{ijk} for each
increasing triple of colors. Each theta_{ijk} is odd:
theta_{ijk}^2 = 0, and theta_{ijk} commutes with theta_{i'j'k'}
when they are distinct (in the super sense: with a sign).
We extend the theta symbols "alternately" to ALL triples of
colors, not just increasing ones, by requiring
theta_{i j k} = alternating in i,j,k, with sign (-1)^{sgn}
theta_{i j k} = 0 whenever two of i,j,k coincide or
the triple is not increasing after sorting.
So for example theta_{2,1,3} = -theta_{1,2,3}, and
theta_{1,1,3} = 0.
The integer g = 10^60 is enormous, so theta_{ijk} has
C(g,3) = D_* independent generators. Any monomial of degree D_*
in theta's is either one specific product
Theta = product_{i < j < k} theta_{ijk}
or has some generator squared and is therefore zero.
The product Theta is THE monomial of maximal degree that is
not automatically zero. It is our eventual "test" for Y.
--------------------------------------------------------------------------------
4.2 The symplectic space
--------------------------------------------------------------------------------
Let W be a purely odd Q-vector space of dimension 2g, with a
nondegenerate symplectic pairing < , >. Choose bases
a_1, ..., a_g (a "Lagrangian" basis)
b_1, ..., b_g (dual basis)
such that
<a_i, b_j> = delta_{ij}, <a_i, a_j> = <b_i, b_j> = 0.
The first g elements a_i span a Lagrangian; the b_j span a
complementary Lagrangian. Both W_+ = span(a_i) and W_- = span(b_j)
are Lagrangian subspaces of W.
W is "odd": we use it inside a super-setting where its elements
are odd. This is why the symplectic form < , >, which is
ordinarily alternating (and hence a "symmetric" form in the
usual sense after the odd shift), can be used as a pairing.
We make an odd alternating three-form Phi on W by decree:
Phi(a_i, a_j, a_k) = theta_{ijk},
Phi(w_1, w_2, w_3) = 0 whenever some w_s lies in W_-.
The value of Phi on a triple of a's is the corresponding
triangle-theta generator. It vanishes whenever any of the three
inputs sits in the b-part.
Since Phi has odd coefficients (the theta's), the ordinary sign
rules make it alternating: Phi(w_2, w_1, w_3) = -Phi(w_1, w_2, w_3),
etc.
--------------------------------------------------------------------------------
4.3 The Frobenius algebra
--------------------------------------------------------------------------------
The state space is the super R-module
V = R (x) ( Q 1 + W + Q e ),
with Q 1 and Q e behaving as a "hyperbolic plane" and W as
discussed above. In words: V is R tensored with the direct sum
of "1, W, e".
Define a trace (or "integral") on V:
int(1) = 0
int(e) = 1
int(w) = 0 for all w in W
int(w_1 w_2) = <w_1, w_2>
int(w_1 w_2 w_3) = Phi(w_1, w_2, w_3)
for w_i in W. Higher-degree products of W's vanish by
convention (see below). The pairing induced by this integral is
the hyperbolic-symplectic pairing: it pairs 1 with e (value 1),
pairs W_+ with W_- (via < , >), and is zero on the tangent
summands (1,1), (e,e).
Define the product by demanding that it match the integral and
the symplectic form and the cubic Phi:
- Let 1 be the identity.
- Let e^2 = 0 and e * W = 0 (so e is a "nilpotent
square").
- For w_1, w_2 in W, define w_1 * w_2 by the rule:
(w_1 * w_2) = <w_1, w_2> * e
+ (the vector in W dual to the functional
w_3 |-> Phi(w_1, w_2, w_3)).
The second term lands in R (x) W_- because Phi has support only
on W_+ in each slot, so its dual vector is automatically in W_-.
Lemma 4.3. These rules give a supercommutative associative
Frobenius algebra. It has an interesting property:
- The product of three standard odd vectors (i.e.
standard w_i in W) is in Re (that is, in R * e, the even
summand);
- The product of four such vectors is zero.
The "handle element", namely the image under mu o Delta of 1,
is
mu Delta (1) = (2 - 2g) e.
Set lambda = (2g - 2) e. Then mu Delta (1) = -lambda.
Proof sketch (full proof in original). The pairing is
nondegenerate, so the products are uniquely determined. Super-
commutativity follows from alternation of < , > and of Phi,
together with the super sign rule.
For associativity, look at the associator of three vectors in
W and pair it with a fourth basis vector. Pairing with 1 gives
cyclic symmetry of the alternating cubic Phi; pairing with e
gives zero. If the fourth vector is in W, both sides become
contractions of two pair-product tensors whose W- and
hyperbolic-channel components vanish by the Lagrangian support
of Phi and by e^2 = eW = 0. Cases involving 1 or e are
immediate by direct computation.
The handle element computation: the hyperbolic channels
contribute +2e, the 2g odd channels contribute -2ge, and the
cubic perturbation has zero internal trace because symplectic
contraction pairs the two different Lagrangians. Summing gives
(2 - 2g) e.
End of Lemma 4.3 sketch.
--------------------------------------------------------------------------------
4.4 Coproduct and the block formula
--------------------------------------------------------------------------------
For a finite nonempty label set L, let Delta_L: V -> V^{(x)L}
be the coproduct -- the adjoint (with respect to the pairing) of
iterated multiplication in the L slots. For L a singleton,
Delta_L is the identity.
The coproduct is coassociative and satisfies the projection
formula. In particular for two-distinct-labels ij:
Delta_{ij}(1) * Delta_{ik}(1) = Delta_{ijk}(1) (4.1)
Delta_{ij}(1)^2 = -Delta_{ij}(lambda) (4.2)
int_j Delta_{ij}(v) = v (4.3)
where "int_j" means the trace in slot j (so the equation is an
identity of tensors with one fewer slot). The minus sign in
(4.2) is important: it matches the negative normal bundle of a
rational-tail boundary divisor. This is one of the places
where geometric intuition guides the algebra.
--------------------------------------------------------------------------------
4.5 The bar transform
--------------------------------------------------------------------------------
The detector will not apply directly to psi and kappa monomials.
Instead, it applies to a different presentation of S_{g,m} in
which the boundary classes are separated out.
For each label i:
psi_bar_i = psi_i - sum_{T >= {i}, |T| >= 2} D_T.
This is psi_i with all rational-tail divisors containing i
subtracted. The result is the "psi at boundary" of i, effectively
the cotangent class restricted to the root after all tails
have been pulled off.
For each x in {1,...,m} in a fixed ordering, let L_x be the
pullback of psi_x from the space defined by the labels 1,...,x:
L_x = psi_x - sum_{T contains x, |T| >= 2,
T\{x} subset {x+1,...,m}} D_T.
For j >= 1:
kappa_bar_j = kappa_j - sum_{x=1}^m (L_x)^j.
The "leading graphless part" of this change of variables is
just psi_bar_i = psi_i and kappa_bar_j = kappa_j - sum_i psi_i^j.
This is an invertible polynomial change of variables, so the
bar monomials form a legitimate basis change in S_{g,m}. The
full change of variables (with all the boundary corrections) is
what lets us cleanly separate "root" and "tail" contributions.
--------------------------------------------------------------------------------
4.6 The detector map t_m
--------------------------------------------------------------------------------
We now define Q-linear maps
t_m: S_{g,m} -> V^{(x_R) m}
for each label set {1,...,m}. The construction is recursive on
m.
Base case (m=0):
t_0(1) = 1 in R
t_0(anything with positive bar-degree) = 0.
In words, t_0 is the augmentation that sends 1 to the unit of R
and everything of positive formal degree to zero.
Induction step. We prescribe t_m on the bar-monomial complement
(i.e., on all monomials that do not involve the boundary tail
divisors). For such a monomial,
t_m( product_i bar_psi_i^{d_i} * product_j bar_kappa_j^{c_j} )
= lambda^{d_1} (x) lambda^{d_2} (x) ... (x) lambda^{d_m}
if all c_j = 0,
= 0 otherwise.
Here lambda lands in the i-th tensor slot. Slots that get no
active input receive a "unit" 1 in V. The values are elements
of V^{x m} via the identifications V = R (x) V, R being the
scalar part.
So t_m sends bar_psi_i to lambda in slot i and bar_kappa_j to
zero in all slots.
For a general m-decorated graph (i.e., involving tail divisors),
t_m applies recursively by:
(a) Discard any graph that is not a "rational-tails graph"
(i.e., not a tree with a single genus-g vertex, the
root, and all other vertices rational). The result is 0.
(b) For each rational descendant subtree with |L| >= 2
active labels at that tail:
- Integrate the decorations on its rational parameter
space (which is Mbar_{0, |L|+1} plus maybe some
forgotten labels) to extract its top-degree integral
(which is a scalar in R).
- Retain this scalar.
(c) At the root (which carries g genus and a slots
corresponding to entries), compute t_a of the root's
decoration with its a active slots, contracting the
already-computed scalars from its rational children
into the tensor product.
(d) Duplicate each root slot into the descendant labels of
its branch by Delta_L. That is, if a rational subtree
carries labels L = {l_1,...,l_k}, apply Delta_L to the
corresponding root slot to distribute the state among
the k descendants.
Every branch attached to the root has at least two
descendant labels (by the definition of rational-tails graph).
This is what makes the recursion terminate: t_m at a root with
a < m labels is a call to t_a with a < m.
Lemma 4.6. The prescription is well defined modulo genuine
(Chow) relations on rational parameter factors, and it induces
a map on S_{g,m}. For a homogeneous expression of degree r, its
image is a sum of tensors
tensor over blocks L of Delta_L(lambda^{d_L}),
indexed by partitions of the label set into nonempty blocks, of
the form
r = sum over blocks L of ( |L| - 1 + d_L ). (4.4)
In particular, t_m = 0 for r > m.
Proof sketch. Grafting a genus-zero graph amounts to composing
coproducts and multiplying integrals; coassociativity makes the
result independent of the internal decomposition. For a rational
subtree with k descendant labels, top decorations plus its
edges going up to the root add to k-1; block-degree bookkeeping
then gives (4.4). Since lambda^2 = 0, a nonzero term has
d_L <= 1 for every block, hence r <= m.
Excessive decorations on a rational vertex have zero integral
by the standard dimension truncation. At a root with a incident
slots, a decoration exceeding 3g - 3 + a also exceeds a
(since g >= 2), and its recursive test is zero. Under
refinement, the sum of vertex dimensions of the pieces decreases
by the number of new edges, so splitting decorations cannot
make an excessive total degree admissible on every piece.
End of Lemma 4.6 sketch.
End of Part 4A. Part 4B (4.7 onward) proves that t_m vanishes
on P_{g,n} and does not vanish on Y.
--------------------------------------------------------------------------------
4.7 Boundary operations
--------------------------------------------------------------------------------
Lemma 4.7. The maps t_m are equivariant for relabeling and
respect the boundary operations of S_{g,m}. Precisely, for every
strata expression b they satisfy
t( psi_bar_i * b ) = lambda_i * t(b) (4.5)
t( kappa_bar_j * b ) = 0 (4.6)
t( D_bar_{ij} * b ) = Delta_{ij}(1) * t(b) (4.7)
where lambda_i denotes lambda in slot i and identity in others.
More generally, gluing along the boundary divisor D_T integrates
the entire genus-zero input and duplicates the root-side
attaching slot into the labels of T.
Proof sketch. The descendant- and kappa-identities are the
forgetting and gluing formulas of Arbarello-Cornalba
(Equations 1.7-1.10). At a gluing D_T, let p be the root-side
flag and p' the rational-side flag. The normal class is
-psi_p - psi_{p'}. On Mbar_{0, T + {p'}}, the sum of boundary
splits separating i from p' equals psi_i + psi_{p'}, and the
sum whose side away from p' contains two fixed labels i,j
equals psi_{p'}. Therefore:
- bar_psi_i restricts to the modified cotangent on the
root side (using p if i in T).
- bar_D_{ij} restricts to the analogous pair class on the
root side, EXCEPT when both labels are in T, in which
case it vanishes.
The bar_kappa_j class is independent of the ordering: for two
consecutive labels made last, D = D_{i,j}, the identities
psi_i|_D = psi_j|_D = 0 and D|_D = -psi_p give, for d >= 1,
(psi_i - D)^d - psi_i^d = -D psi_p^{d-1}
= (psi_j - D)^d - psi_j^d.
This is exactly the invariance under a consecutive transposition
that proves well-definedness of kappa_bar_j as a bar-class.
Restricting kappa_bar_j to D_T: put the labels of T last; the
kappa class splits additively; the first label of T gives
root-side cotangent at p; the remaining partial cotangents on
the rational factor cancel its kappa class by the forgetful
kappa formula. So kappa_bar_j restricts to itself on the root
side. Combined with the coproduct projection formula, this
proves (4.5)-(4.6).
For (4.7): only the two-label tail has nonzero undecorated
integral. If i,j sit in different branches, the root-side pair
class joins their ancestor slots, giving Delta_{ij}(1). If they
sit in the same branch, its restriction is -psi_p; the identity
(4.2) turns this into Delta_{ij}(1) via the self-intersection
(delta_{ij}(1))^2 = -delta_{ij}(lambda).
These calculations take place in the auxiliary formal graph
space modulo relations on rational parameter factors.
Simultaneous compatible splits produce the same tree
independently of order; polynomial multiplication commutes and
associates. A genuine zero on a rational factor remains zero
under further operations. Lemma 4.6 permits the dimension
truncations, since discarded terms remain outside the
rational-tails locus under refinement.
End of Lemma 4.7 sketch.
--------------------------------------------------------------------------------
4.8 Forgetting operations
--------------------------------------------------------------------------------
Lemma 4.8. For every strata expression b, the maps t_m satisfy
t( pi_y^* b ) = t(b) (x) 1 (4.8)
t( pi_{y,*} b ) = int_y t(b) (4.9)
where pi_y forgets the label y and int_y is the trace in the
slot at y. Identities hold in any order of the remaining labels.
Proof sketch. Both sides are Q-linear in b; it suffices to check
on the bar-monomial complement.
Forget pullback on a vertex: replaces
psi_h |-> psi_h - D_{h,y}
kappa_j |-> kappa_j - psi_y^j.
A tail divisor D_T pulls back to D_T + D_{T + {y}}.
The key observation: the (formal) pullback commutes with
multiplication by the indicated polynomial classes, because
incompatible splits stay incompatible, and compatible splits
either split a vertex (a vertex calculation) or add a boundary
term cutting off the half-edge at y together with the branch.
Pushforward at a vertex: put X_i = pi_y^* psi_i, k_j = pi_y^* kappa_j,
d_i = D_{i,y}. Then psi_i = X_i + d_i, kappa_j = k_j + psi_y^j,
and the local rules are
d_i d_l = 0 (i != l), d_i^2 = -X_i d_i, psi_y d_i = 0
pi_{y,*} 1 = 0
pi_{y,*} psi_y^d = kappa_{d-1} (d >= 1)
pi_{y,*} d_i = 1
The projection formula for graphless cotangent and kappa
polynomials follows by substitution. For a tail multiplier, the
two pullback splits distribute kappa_{d-1} additively; for d_i,
only the compatible split pushes to the original tail. On a
tree with a stable y-vertex, this reduces to the same vertex
calculation.
The ordinary bar-monomial pushforward uses the telescoping
identity
psi_y^d - psi_bar_y^d
= sum_{S nonempty} D_{S + {y}} sum_{a=0}^{d-1}
psi_y^{d-1-a} psi_bar_y^a
and the symmetric-power integral
int_{Mbar_{0, |S|+2}} psi_y^{|S|-1} = 1.
For d=1 the scalar identity kappa_0 = 2g-2+m applies; for d>1,
expand kappa_{d-1} = kappa_bar_{d-1} + sum_x L_x^{d-1} and
compute similarly. Both give
t_m( pi_{y,*} psi_bar_y^d ) = int(lambda) = 1 in R.
The compatibility with the cutoff (Lemma 4.6): an excessive
decoration on a rational factor is a genuine zero in its
genus-zero Chow ring, hence has zero forget pullback or
pushforward. For an excessive root decoration of degree d with
a root slots (so d >= 3g-2+a), adding a mark brings total
degree d > a+1, killed by the block bound with a+1 slots;
forgetting a mark brings degree d-1 > a-1, killed by the block
bound with a-1 slots. The root stays stable since g >= 2.
Equivariance from Lemma 4.7 lets the forgotten label be placed
last in any chosen order.
End of Lemma 4.8 sketch.
--------------------------------------------------------------------------------
4.9 The zero-leg identity
--------------------------------------------------------------------------------
We now need to know how t_m treats a Pixton relation with one
leg of index zero (that is, a_i = 0 for one i). This requires
a technical identity.
Package the partition insertions with formal variables s_l,
one for each allowed positive part, using exponential
coefficients for repeated parts, and define
H(T, zeta) = Chat_0(T, zeta) - sum_l s_l Chat_l(T, zeta).
Then a cycle-exponential formula for kappaHat_Gamma rewrites
the vertex characteristic as
exp{ -log H }_{kappa, zeta_v} (4.10)
where a monomial T^d zeta^epsilon in -log H becomes
kappa_d T^d zeta_v^epsilon, including d=0. This formula is
exactly the standard identity that lets kappa-operators be
assembled from elementary vertex factors.
For the edge we use U(x,x') = T Delta_e, so that the coefficient
of total degree T^r is the r-th coefficient of the raw recipe.
Denote the packaged series by R(s; a).
Lemma 4.9. If all preceding leg indices a_j are in {0,1},
then, in the untruncated rational-tails computation,
R(s; a, 0) - sum_l s_l R(s; a, l)
= pi_y^* R(s; a). (4.11)
Proof sketch. The kappa-pullback formula, including
kappa_0^{old} = kappa_0^{new} - 1, multiplies the new
characteristic (4.10) by H(psi_y T, zeta_v). This is exactly
the linear combination of new leg factors on the left.
The remaining check is bubbles introduced in pulling the old
leg and half-edge cotangents. For a function f(x), that pull
contributes a boundary term (f(0) - f(x))/x, where x is the
stable-side cotangent. At a new rational three-point vertex,
the vertex characteristic is H(0, epsilon)^{-1}; the new
H-leg cancels it. Writing A_x = A(eta x T), etc., the edge
adjoining such a bubble contributes
[eta (B_x + 1) + epsilon (1 - A_x)] / x (4.12)
If the bubble carries an old leg, its constant factor is 1 or
-epsilon. Extracting epsilon gives (1 - A_x)/x or
-eta(B_x + 1)/x, exactly the divided differences needed. If it
subdivides an old edge, the epsilon-coefficient of the product
of two edge factors is
[eta(B_x+1)(1-A_{x'}) + eta'(B_{x'}+1)(1-A_x)] / (x x') (4.13)
which equals the difference formula
U(0,x')-U(x,x') over x plus U(x,0)-U(x,x') over x', giving
both cotangent pullback corrections. Bubbles for distinct
slots at a vertex are disjoint, and these are all possibilities
for adding one mark. Rational-tails graphs have h^1=0 and
trivial automorphisms, so neither the 2^{-h^1} nor the
unnormalized gluing introduces another coefficient.
End of Lemma 4.9 sketch.
--------------------------------------------------------------------------------
4.10 Annihilating the Pixton span
--------------------------------------------------------------------------------
Lemma 4.10. For every m >= 0, t_m( P_{g,m} ) = 0.
Proof sketch. We induct on paired data: first on arity m, then
within fixed arity on (r, ell(sigma)) lexicographically.
Smaller-arity basis. At m=0, every admissible basic Pixton
relation has positive degree, while t_0 kills all positive
degrees.
Nontrivial outer gluing. If the outer graph is not a
rational-tails graph, all its refinements have zero test. If
it is, a relation at a rational factor has zero top integral
by Chow validity. A relation at the root is killed by the
smaller-arity induction (the root has fewer labels). Integration
and coproducts then give zero for the entire generator.
Multiplication by a tail divisor. If b in P_{g,m}, then by the
ideal property and (4.7) applied to the D_T-multiplied Pixton
relation, t_m(D_T b) = 0 for any T. This identity uses only
smaller-arity induction on the root factor.
Basic recipe relations. This is the main case. Using the
substitution psi_i = psi_bar_i + sum_T D_T:
- The all-bar-psi term uses the lower triple (m, r-k, ell(sigma))
and is killed by (4.5)-(4.6).
- Other terms are D_T b' with b' in P_{g,m}, killed by the
tail-divisor ideal step above.
So we may assume all leg indices are in {0,1}.
Then we may assume r <= m (Lemma 4.6). If a leg has index zero,
apply Lemma 4.9. The pullback term reduces to (m-1, r, ell(sigma))
and dies by induction on arity. The "move a part to the zero
leg" term strictly decreases ell(sigma) or r. Reducing a
positive part ell = 3k+h (h in {0,1}) first, and reapplying the
above moves, maintains the well-founded induction.
All-ones case. The Pixton recipe at all-ones is symmetric
under label permutations, and so is its test tensor. Contracting
any one slot with 1 kills it (a forget pushforward), and
contracting with e also kills it (multiply by psi_bar_y/(2g-2)
first, then forget). So we need only test on standard W vectors.
Block counting. In the block-expansion (4.4), a singleton has
zero pairing with an odd input, a positive power of lambda
kills odd inputs, and a block of >= 4 odd inputs has zero
product. So only undecorated pair and triple blocks survive:
with s pairs and u triples,
m = 2s + 3u, r = s + 2u, s + 3u = 3r - m >= g + 1 + |sigma| > g.
Every alternated term uses more than g coordinates from the
g-dimensional W_+ dual and is zero. All standard input tests
vanish; nondegeneracy of the pairing gives the tensor itself
zero.
End of Lemma 4.10 sketch.
--------------------------------------------------------------------------------
4.11 Nonzero on Y
--------------------------------------------------------------------------------
We finish Claim A by showing t_n(Y) is nonzero.
Every graph in a bridge correction beta_J^st has nontrivial
unmarked stabilization, hence is not a rational-tails graph.
Its refinements, pullbacks, and products remain outside the
support of the test. Therefore, by (4.5)-(4.6) and (4.1), the
remaining part of each factor of Y tests as
Delta_{ab}(1) Delta_{ac}(1) = Delta_{abc}(1).
Pair slot j with a_{i_j}, the W_+ basis vector prescribed by
the color word. A block with increasing colors i < j < k gives
theta_{ijk}, and the product of unsymmetrized factors
evaluates to
Theta = product_{i < j < k} theta_{ijk},
up to one fixed super sign.
Let G_i be the color-word stabilizer, from Definition 3.4. For
tau in G_i, transporting its super permutation across the
pairing of odd slots contributes sgn(tau); this cancels the
explicit sign of that summand in Y. No reordering of the
exterior generators is required: all summands are compared
with the same ordered evaluation. Every color occurs C(g-1,2)
times, hence
< t_n(Y), a_{i_1} (x) ... (x) a_{i_n} >
= +/- ( C(g-1,2)! )^g * Theta
!= 0. (4.14)
Combined with Lemma 4.10, this gives t_n(Y) != 0, and therefore
Y is not in P_{g,n}. That proves Claim A.
--------------------------------------------------------------------------------
4.12 The formal detector stated
--------------------------------------------------------------------------------
Proposition 4.11. There exist a supercommutative Q-algebra R,
a finite free supermodule V over R, and Q-linear maps
t_m: S_{g,m} -> V^{x m} (m >= 0)
such that t_m( P_{g,m} ) = 0 for every m, whereas t_n(Y) != 0
for the class Y of Definition 3.4 and the n of Theorem 2.1.
In particular, Y is not in P_{g,n}.
This Proposition IS Claim A of Theorem 2.1. The proof is the
combination of Parts 4.1-4.11 above.
The maps t_m are tests on FORMAL strata: they are not asserted
to factor through the Chow ring. This is a pure S_{g,m}-statement.
The geometric witnesses they provide are: Theta for the color
product, lambda^{d_i} for the psi decorations, and Delta_L for
the branch structure.
End of Part 4. Part 5 begins Claim B: proving q_{g,n}(Y) = 0.
=====================================================
PART 5: WHY Y VANISHES IN CHOW
=====================================================
We now turn to Claim B: q_{g,n}(Y) = 0 in A^*(Mbar_{g,n}; Q).
This is by far the harder half of the proof. Part 5 sets up
the geometric machinery; Part 6 handles the boundary.
The strategy: attack q(Y) from two directions at once.
Direction 1 (smooth families). On a family of smooth curves,
Y restricts to a specific class built from "small diagonals".
We can prove a bound on powers of these small diagonals using
Quot schemes on the surface C x C.
Direction 2 (compactification). On the boundary, Y restricts
to a "corrected cubic" that differs from the smooth-curve
class by controlled corrections. We prove a support-dimension
induction that pushes Y to vanish component by component.
The two directions fit together in Part 6. Part 5 handles
Direction 1: the surface relation.
--------------------------------------------------------------------------------
5.1 The faithful Chow tensor model
--------------------------------------------------------------------------------
Let pi: C -> S be a smooth projective family of connected curves
of genus b over an irreducible smooth quasi-projective complex
variety S, with a section. Fix a rational divisor class
e_C in A^1(C) of relative degree one. The class e_C need not
be the class of the section.
Set:
a_C = pi_*(e_C^2) (a base class)
e'_C = e_C - (1/2) pi^*(a_C) (a "corrected degree-one")
eta = [Delta_{C/S}] - e'_{C,1} - e'_{C,2} (the "projector")
Here Delta_{C/S} is the diagonal, and e'_{C,i} means e'_C pulled
back via i-th projection of C x_S C.
The point of the correction: e'_C is a degree-one class whose
self-intersection lands in the base (not in C). Combined with
the base-section algebra, this gives a decomposition of C into
a "hyperbolic plane" (unit and correction) plus a leftover
summand. The projector eta isolates that leftover summand: it
is a self-transpose (up to sign, in the super sense) idempotent
whose trace equals -2b.
This is the standard tensor-model decomposition used all over
the tautological literature (Gross-Schoen, O'Sullivan). Its
whole reason to exist is that the surface Quot relation of
Section 5.2 can be stated cleanly in it.
The category we work in is the rational correspondence
category Ch with:
- objects: smooth projective S-families (of curves, curve
powers, etc.);
- morphisms Hom(P,Q) = A^*(P x_S Q);
- composition: pull to triple product, intersect, push;
- tensor: relative product;
- finite direct sums and split idempotents allowed.
We may quotient by classes that vanish on S minus a codimension-a-
subset; call this "the base-support quotient." It is not
numerical equivalence, and it is not testing individual fibers.
Proposition 5.1 (Faithful Chow tensor test). In this setup there
is a supercommutative rational algebra B_b with:
(i) A faithful tensor model of the curve algebra on
B_b (x) (Q 1 + W_b + Q e), where W_b is an odd
symplectic space of dimension 2b. The classes
1_C and e'_C correspond to 1 and e; the projector
eta projects onto W_b. Integration gives the hyperbolic
pairing <1,e>=1 on the even summands and the symplectic
pairing on W_b. All actual Chow correspondences between
curve powers are faithful in this model, not only
pairing diagrams.
(ii) The projected small diagonal is an alternating three-
form phi: (W_b)^3 -> (B_b)_odd. Cup product, tensor,
permutation, contraction are computed with Koszul signs.
(iii) For an ordinary rational color space U: a pure colored
tensor in
( eta^{(x)k} A^*(C^k) (x) U^{(x)k} (x) sgn_k )_{S_k}
is zero iff all its evaluations using the same ordinary
linear map z: U -> W_b in every slot are zero in B_b.
Finitely many rational maps z suffice for each fixed U,k.
The same holds in the base-support quotient.
The most important clause here is faithfulness: this is not just
an analog model. It is an actual isomorphism onto the Chow
classes. In particular, any class the model cannot see has some
specific evaluation into B_b vanishing, rather than some analogy
failing.
For a pointed curve over a field, the projected small diagonal
is the modified diagonal of Gross-Schoen. In a family, the base
corrections (the (1/2) pi^*(a_C) terms) are part of the same
construction. This is exactly what F_J^st - (1/2) beta_J^st is
modeling at the compactified level.
--------------------------------------------------------------------------------
5.2 The surface Quot relation
--------------------------------------------------------------------------------
Now suppose we have, in addition, a smooth family of curves of
genus b over a smooth base S, as in Proposition 5.1. We form the
surface
X = C x_S C,
pi_X: X -> S,
K = K_{X/S}.
Every fiber of X is a surface with K ample, K^2 = s_K = 8 chi,
Euler characteristic chi = (b-1)^2.
Definition 5.2. Let Quot be the relative Quot scheme of
rank-zero quotients
0 -> I' -> V -> Q -> 0
where V = L_1 + L_2 = K + O_X, subject to:
D = d K = 3K,
det Q = D in Pic(X/S) (a determinant constraint),
int_{X_s} ch_2(Q_s) = nu + L_1 D - D^2/2,
nu = t_0 - 1.
Here t_0 = 25 chi - 1 (this is the t_0 from Theorem 5.3 below).
The numerical parameters:
d = 3, s_K = 8 chi, chi = (b-1)^2
t_0 = chi + d(d-1)/2 * s_K - 1 = 25 chi - 1
m = 3 t_0.
Quot is a projective moduli space over S, by Nitsure's Quot
theorem plus the Kleiman Picard scheme for the determinant
constraint.
The key structural facts about Quot that we need, in order:
(a) Derived obstruction theory. The determinant constraint
changes both the obstruction space and the virtual
dimension. The map to based Picard (via trace of
determinant) is surjective on Ext^2, giving a two-term
perfect obstruction theory. This is a standard application
of the perfect-complex moduli theory (TV, STV).
(b) Fixed loci and virtual classes. The torus acts with fixed
subsheaves splitting into the two weight components. Each
rank-one factor has a unique "divisor + residual point"
factorization (Fogarty). Setting up a low/high index
choice lets us express the fixed virtual class as a
virtual pullback from a low-side Quot scheme, which
reduces to a product of a divisor scheme with a Hilbert
scheme of points.
(c) Marked localization. We attach m = 3 t_0 independent
surface positions, insert ch_2^{T}(Q) at each, and take
the localized pushforward to X^m. Tracking the codimension
budget shows only a specific low-index branch survives,
and only its bare-triple partitions contribute a nonzero
scalar.
(d) Marked universality (Lemma, EGL relative version).
Pushing forward finitely many universal point-ideal
Chern characters (with marked surface outputs retained)
gives a universal combination of decorated equality
diagonals. This lets us identify the surviving scalar
without committing to a fiberwise computation first.
The upshot is the localization relation in the following form:
Theorem 5.3 (Surface bound, Chow). Suppose b >= 5 and
dim S < chi - 1 = (b-1)^2 - 1. Let (w_alpha)_{alpha in I} be a
finite collection of vectors in the odd symplectic standard
space W_b with isotropic span. For any matrix of scalars
gamma = (gamma_{alpha beta}), define
T_gamma = sum over alpha_1,alpha_2,alpha_3
beta_1,beta_2,beta_3
( product_{j=1}^3 gamma_{alpha_j beta_j} )
phi(w_{alpha_1},w_{alpha_2},w_{alpha_3})
phi(w_{beta_1},w_{beta_2},w_{beta_3}).
Then
T_gamma^{t_0} = 0 in B_b.
The assertion remains valid after any characteristic-zero scalar
extension.
Interpretation. The right object to think about is not "small
diagonals" but "products of two cubic phi values tied by the
matrix gamma". If we plug in isotropic vectors w_alpha -- that is,
vectors whose symplectic pairings vanish on the span -- the
surface relation says the t_0-th power of that product is zero.
This is the "surface Quot relation" of the paper. It is the
geometric axiom that produces the boundedness we need in
Part 6.
--------------------------------------------------------------------------------
5.3 The scalar a_{N,t_0} is nonzero
--------------------------------------------------------------------------------
The proof of Theorem 5.3 has a scalar factor that could in
principle vanish. The next result shows it doesn't.
Proposition 5.4 (Lehn-type scalar). With the parameters of
Theorem 5.3, the scalar a_{N,t_0} produced by the marked
localization computation is a nonzero rational number.
The scalar a_{N,t_0} is defined by a residue:
a_{N,t_0}
= (1 / (6^{t_0} t_0!))
[q^{d s_K - 1}]
(1 + w_0)^{d s_K} (1 + 2q)^{t_0} (1 + 4q)^E
where w_0(1 + w_0) = q with w_0(0) = 0, d = 3, s_K = 8 chi,
t_0 = 25 chi - 1, chi = (b-1)^2, and
E = d(d+1) s_K / 4 - 3 t_0 / 2.
We won't reproduce the whole derivation here -- it lives in
Sections 5-6 of the original paper, a difficult Hilbert-scheme
interpretation. What matters is the conclusion: substitution
v = 1 + 2 w_0 turns the coefficient into a residue at v = 1,
Res_{v=1} [ (1 + v^2)^{t_0} v^{2E+1} ] / (v-1)^{d s_K} * dv.
This residue computes (up to sign) the residue at v = 0:
[v^{27 chi - 5}] (1 + v^2)^{25 chi - 1} (1 - v)^{-24 chi}.
The v = 0 residue is nonzero and positive, because:
- the factor (1 + v^2)^{25 chi - 1} has nonnegative coefficients
and constant term 1;
- the factor (1 - v)^{-24 chi} has strictly positive coefficients
in every nonnegative degree;
- the exponent 27 chi - 5 >= 0.
Hence the v = 1 residue is a negative scalar multiple of a
positive integer and is nonzero.
The exact bookkeeping of prefactors 2^{d s_K - t_0 - 1} and
1/(6^{t_0} t_0!) is nonzero too. Combining: a_{N,t_0} != 0.
Consequently Theorem 5.3 holds: since a_{N,t_0} is invertible
in B_b, T_gamma^{t_0} = 0 follows from a_{N,t_0} T_gamma^{t_0} = 0.
--------------------------------------------------------------------------------
5.4 Arbitrary color products
--------------------------------------------------------------------------------
Theorem 5.3 gives nilpotence only for products whose inputs are
isotropic vectors in W_b. That's a harsh restriction: our
classes Y involve arbitrary color inputs, not isotropic vectors.
To remove the restriction, we prove a purely linear-algebra
lemma.
Lemma 5.5 (An exterior ideal). Let F subset H be vector spaces
of dimensions 2r and D, let Omega in Lambda^2 F be nondegenerate
(a symplectic form on F). If 0 <= t <= r, then the ideal in
Lambda H generated by Omega^t contains Lambda^q H for every
q >= D - r + t.
Proof sketch. The linear symplectic Lefschetz decomposition
gives surjectivity
Omega^t ^ (-): Lambda^{k - 2t} F -> Lambda^k F
for k >= r + t. Split H = F + F'. A nonzero summand of Lambda^q H
has F'-degree at most D - 2r, hence F-degree at least
q - (D - 2r) >= r + t. That summand is in the ideal Omega^t
(Lambda H). So the whole Lambda^q H is in the ideal.
Interpretation: this says a large positive power of the
symplectic form kills everything of high enough exterior degree.
"Large enough" is D - r + t.
Now the application to colors. Take an m-dimensional color space
E. Fix an even integer e with e >= m/2 - 2 and e <= m/2. Write
D_E = C(m,3), d_e = C(e,3).
Lemma 5.6 (Arbitrary color products). Let a smooth curve family
of genus b >= 5, with a section, be over a smooth irreducible
quasi-projective base S satisfying dim S < (b-1)^2 - 1 = chi - 1.
Put
t_b = 25 (b-1)^2 - 1 = 25 chi - 1 = t_0.
If t_b <= d_e/2, then every product of
q >= D_E - d_e/2 + t_b
pure cubic entries whose colors lie in E vanishes modulo
support in base codimension at least b-1.
Proof sketch. Fix a color test z: E -> W_b. The pullback of the
symplectic form to E has a totally isotropic subspace of
dimension at least m/2: take the radical plus a Lagrangian in the
symplectic quotient.
Choose an e-dimensional subspace E' of it. Its image under z is
isotropic. Choose an invertible skew matrix gamma in a basis of
E'. Grouping the ordered triples of E' into a quadratic expression
in cubic generators, the coefficients of the resulting quadratic
form are a nonzero scalar times the 3x3 minors of gamma, hence a
nondegenerate bivector
Omega in Lambda^2 ( Lambda^3 E' ).
Theorem 5.3 says (via a specialization) that Omega^{t_b} kills
the whole cubic coefficient space in the specified support
quotient.
Lemma 5.5 then says: with F = Lambda^3 E' (dimension 2r where
2r = d_e, and r = d_e/2), H = Lambda^3 E (dimension D_E), any
exterior element of degree >= D_E - d_e/2 + t_b is in the ideal
Omega^{t_b}. This is exactly the threshold in Lemma 5.6.
Apply Proposition 5.1 (faithfulness). All color maps z agree on
the threshold, so the vanishing is uniform in z. Faithful
polarization (Prop 5.1(iii)) upgrades this to a vanishing in
Chow, modulo the base-support bound.
The base-support bound b-1 is required everywhere: it is the
quotient by which we already lost information when we restricted
to the surface.
--------------------------------------------------------------------------------
5.5 What Part 5 has given us
--------------------------------------------------------------------------------
We now have:
(i) A faithful model of Chow correspondences on curve powers
in a coefficient algebra B_b with an odd symplectic W_b
and a projected cubic phi.
(ii) A surface Quot relation: T_gamma^{t_0} = 0 for
isotropic-gamma contractions, t_0 = 25 (b-1)^2 - 1.
(iii) The scalar a_{N,t_0} in the surface relation is
nonzero.
(iv) An exterior-ideal argument: arbitrary color products
of >= D_E - d_e/2 + t_b pure cubic entries vanish
modulo base codimension b-1.
What's missing, and what Part 6 supplies:
(v) The compactification step: the class Y, restricted to
the boundary, agrees with a corrected cubic that we can
compare to the pure cubic entries from (iv), up to
controlled errors over a dense open.
(vi) Support induction: iteratively reduce the dimension of
the base-image of the residual loca, using (v),
until the support is empty.
Part 6 does (v) and (vi). That completes Claim B and the proof
of Theorem 2.1.
End of Part 5. Part 6 handles the compactification.
=====================================================
PART 6: COMPACTIFICATION AND THE LAST STRETCH
=====================================================
Part 5 proved a bound for pure cubic entries over smooth curve
families. Part 6 lifts this to the entire compactified moduli
space Mbar_{g,n}. Two things happen here:
1. Boundary analysis: we build a "corrected cubic" class
F^0_{h,B} on weighted compactifications whose restriction
to the smooth core agrees with the projected small
diagonal (from Part 5).
2. Support induction: on the boundary, F^0_{h,B} differs from
the smooth-curve class by controlled errors. We iteratively
lower the dimension of the base image of the residual
cycle until it is empty.
The key geometric objects are Hassett's weighted stable curves
with light and heavy markings. We use these because plain
Mbar_{g,n} is not fine enough for the comparison: we need to
allow lights to collide while heavy marks stay separate.
The chapter is long. The overall shape:
6.1-6.2 Weighted conventions and the corrected cubic
6.3 The blowup at a triple flag: three-position
calculation
6.4 Pure lifts: F^0 controlled by G_h,B
6.5 Support advancement lemma (dimension induction)
6.6 Finite color budget (arithmetic)
6.7 Wrap
--------------------------------------------------------------------------------
6.1 Weighted configurations and cut charts
--------------------------------------------------------------------------------
Let (h,B) be "heavy data": B is a finite ordered set (the heavy
marks) and 2h-2+|B| > 0 (heavy stability). Write
X_{h,B}(k) = Mbar_{h, (1^B, epsilon^k)} for 0 < k epsilon <= 1,
X_{h,B}(0) = Mbar_{h,B}.
The first argument is the genus; the entries in (1^B, epsilon^k)
specify heavy weights (all equal 1) and light weights (all equal
epsilon, small and positive, total <= 1).
Hassett's theory: for any positive rational weights with total
at most 1, the moduli space of weighted pointed stable curves
is a smooth proper Deligne-Mumford stack. Lights can collide
with one another (but not with heavy marks); one-light space is
the universal curve; weight reduction gives forgetting push and
pull maps.
A rational component with one node and only light markings is
forbidden by stability. A rational component with two nodes and
positive light mass is allowed and stable.
Now fix a stable graph Gamma of the heavy type. At a vertex v
with genus h_v and heavy set B_v (original marked points plus
incident half-edges, all ordered), write
B_Gamma = product_v Mbar_{h_v, B_v} (heavy base)
S_Gamma = product_v M_{h_v, B_v} (smooth core)
The cut chart with k lights:
X^Gamma(k) = coproduct over assignments a: {1,...,k} -> V(Gamma)
product_v X_{h_v,B_v}( a^{-1}(v) ).
So a cut chart records how k lights are distributed among the
vertices. It has a proper gluing map to X_{h,B}(k). Assignment
conditions are part of the indexing: a class supported on a
specific assignment is zero on the other components.
Cutting and light forgetting commute: the heavy flag replacing
a node branch retains its stability contribution, so stabilizing
after gluing gives the same curve as forgetting-then-stabilizing.
We work in rational Chow with operational (bivariant) behavior.
For an ordinary color space U, the operation algebra we use is
(+)_{k >= 0} ( A^*(X^Gamma(k)) (x) U^{(x)k} (x) sgn_k )_{S_k}.
The product pulls operations to disjoint sets of labels by
forgetting and then multiplies. Cubic and singleton operations
are odd, pair operations are even. All expressions below are
first identities of labeled classes, then identities in this
algebra.
--------------------------------------------------------------------------------
6.2 The corrected cubic F^0_{h,B}
--------------------------------------------------------------------------------
For three light labels 1,2,3, define
F^0_{h,B} = [1=2=3]
- (1/2) beta_{h,B}
+ (1/2) * sum_{p in B} delta_{p,123} psi_p (*)
in A^2(X_{h,B}(3)). Each term:
- [1=2=3]: coincidence class, product of [1=2] and [1=3].
- beta_{h,B}: take each stable one-edge graph of the heavy type,
subdivide its edge by a genus-zero vertex carrying exactly
the three lights 1,2,3, sum the resulting gluing classes
divided by their graph automorphism orders. This is the
"rational bridge" correction on the weighted compactification.
- delta_{p,123} psi_p: for p in B (heavy mark), the divisor
where p and 1,2,3 sit on a rational tail; psi_p is the heavy
cotangent class on the weighted space.
All terms are symmetric in the three light labels. This is
F^0 ("raw cubic"). It is NOT the final smooth pure diagonal --
that gets built below.
Lemma 6.1 (Weighted pullback). Let g >= 2 and heavy set B empty.
Pull three-light operations to any larger number of light labels
by forgetting. Under weight reduction from ordinary stable
pointed curves:
[i=j] pulls back to D_bar_{ij},
beta_{g,empty} pulls back to the normalized two-edge bridge sum
used to define beta_J^st.
Consequently the coefficient of the prescribed color word in the
signed product of all F^0_{g,empty} pulls back to q(Y), up to a
nonzero rational scalar.
Proof sketch. In codimension one, the inverse image of [i=j] is
exactly the rational tails containing both labels. At the generic
point of each tail, the two coarse positions differ by a
node-smoothing parameter, and every multiplicity is one.
For the bridge: both bounding nodes exist already in the source
(weight reduction does not contract a bridge with positive light
mass). Therefore the inverse image is supported on the
corresponding ordinary strata of codimension at least two. At a
generic two-edge stratum the three labels are distinct on the
bridge and no component is contracted, so the multiplicity is
one. Further collisions add edges and have higher codimension.
Forgetting commutes with reduction. Extracting a fixed color
word from signed averaging retains exactly the permutations
preserving that word, with signs. Depending on averaging
convention there is a positive factorial denominator, but no
zero scalar.
End of Lemma 6.1 sketch.
--------------------------------------------------------------------------------
6.3 The local three-position calculation
--------------------------------------------------------------------------------
Over the smooth core S_Gamma, let C_v denote the smooth curve at
vertex v. For a fixed assignment of three light labels, there is
a coarse position map
c: X^Gamma(3)|_{S_Gamma} -> C_{a(1)} x_{S_Gamma}
C_{a(2)} x_{S_Gamma}
C_{a(3)}
(all products over S_Gamma).
Lemma 6.2 (Three-position blowup). The map c is the iterated
blowup obtained, at each flag p, by:
- first blowing up the locus where all three assigned
positions equal p (when present);
- then blowing up the strict transforms of the loci where two
assigned positions equal p.
The latter centers are disjoint after the first blowup.
Proof sketch. Extra components above a smooth pointed core are
chains attached at its flags. Write a local parameter x_i for
each label's coarse coordinate near a flag. The zero divisor of
x_i is the sum, with multiplicity one, of the cuts moving label
i onto the expanded chain. Cutting and smoothing satisfy
x*y = t, so at a fixed point the cuts form an initial segment
of the ordered chain. Each relevant pair/triple ideal is
principal after pullback, giving the first blowup then the
successive pair blowups.
Blowup ratios recover the expanded configuration: on a level,
the leading ratios of coarse-coordinate differences are the
ratios of light coordinates on that level, modulo common scaling.
Labels on separate levels give zero/infinity. Forgetting to
pairs gives all these ratios compatibly, so the ratios determine
levels and their configurations.
Automorphisms fixing the core and marked lights are trivial. The
resulting representable map is quasi-finite, proper, birational
to a smooth target [Stacks 02LS and 0AB1]. Working on an etale
chart of the core data, it is an isomorphism.
In the first exceptional plane, the strict pair centers meet at
three distinct coordinate points, proving their disjointness.
Constructions at distinct flags have disjoint centers.
End of Lemma 6.2 sketch.
Denote the total first exceptional divisor at a triple flag
locus by E_{p,123}, and pair exceptional divisors (preferably
pulled back from two-light spaces) by E_{p,ij}.
The blowup formula for Chow groups (Fulton Prop 6.7(e)) gives
that any codimension-two class on c has, in addition to its main
pullback, terms:
E_{p,ij} A_{p,ij} + E_{p,123} D_p + a_p E_{p,123}^2 (**)
A_{p,ij} a divisor class on the remaining curve over the base;
D_p a base divisor class;
a_p a scalar on each connected component of the base.
If we work with the STRICT pair-exceptional divisor E_{p,ij}^str
versus the PULLED BACK pair-exceptional divisor E_{p,ij}^pull,
then
E_{p,ij}^pull = E_{p,ij}^str + E_{p,123}.
Multiplying by a curve divisor D_{p,ij}, this changes only the
D_p term, not the a_p E_{p,123}^2 coefficient. So using pulled
pairs is harmless.
When a raw class F^0 is pulled from an ancestor chart, its main
component is the small diagonal if all three labels are assigned
to one relevant descendant vertex, and zero otherwise. The
coincidence/bridge/heavy-tail corrections project to loci where
all three coarse positions are flags, of codimension three; they
cannot alter a main codimension-two class. Hence the divisor
coefficients A_{p,ij} of F^0 are supported on flag sections of
the remaining curve, up to base terms.
Lemma 6.3 (Triple-exceptional cancellation). For the pullback
of a raw cubic F^0 from an ancestor chart, the coefficient a_p
in (**) is zero at every triple flag locus.
Proof sketch. Over a field point of the smooth heavy base, the
full fiber over three positions at a flag is the weighted chain
space with two heavy endpoints and three lights: the Losev-Manim
space. By Lemma 6.2 it is the first exceptional P^2 blown up at
its three pair-collision points.
The total class E_{p,123} restricts to the pullback of
O_{P^2}(-1); hence
int E_{p,123}^2 = 1.
Main pullbacks, pair terms times remaining-position divisors, and
base terms have integral zero on this fiber. So this integral
detects exactly a_p.
The coincidence term gives 1: its locus is the single transverse
configuration where all three lights coincide away from endpoints.
If p is a NEW cut flag (from a fuller graph), there is one pair
of bounding nodes isolating the chain carrying the three lights.
With these cuts distinguished, the bridge locus is a regular
two-node condition in a versal deformation chart. Its two
normal lines are tensor products of branch tangent lines; the
core factors are constant, so its self-intersection restricts to
(-psi_0)(-psi_infinity) = psi_0 psi_infinity.
The normalized bridge sum counts this local branch once (this
also handles self-nodes).
Reduction from Mbar_{0,5} to the chain space preserves both
heavy cotangent lines. Genus-zero string recurrence gives
int_{Mbar_{0,5}} psi_0 psi_infinity
= int_{Mbar_{0,4}} psi_0 + int_{Mbar_{0,4}} psi_infinity
= 2.
Hence the bridge contribution to F^0 is -1/2 * 2 = -1, canceling
coincidence +1.
If p is an INHERITED heavy flag, the relevant term is the unique
outer tail containing that heavy mark and all three lights. Its
normal factor is -psi_0 on the chain; the inserted psi_p is
psi_infinity. Thus delta_{p,123} psi_p has integral -2, and its
coefficient +1/2 again cancels the coincidence term.
No other bridge or heavy-tail type meets this fiber: after lights
are forgotten, only the specified node or specified heavy mark
survives.
So a_p = 0 on every connected smooth-core base component.
End of Lemma 6.3 sketch.
--------------------------------------------------------------------------------
6.4 Controlled pure lifts
--------------------------------------------------------------------------------
For a smooth heavy core pi: C -> S, choose a degree-one rational
class
e_C = [p] if a heavy flag p is chosen,
= c_1(omega_{C/S}) / (2h-2) if B is empty (h >= 2).
Then put
e'_C = e_C - (1/2) pi^* pi_*(e_C^2)
eta_C = [Delta_C] - e'_{C,1} - e'_{C,2}
Delta_C^pure = small diagonal projected by eta_C in each of its three slots.
The expansion
[Delta_{123}] - Delta_C^pure = (sum of pair-singleton products
with singleton factors 1_C or e'_C)
is a factual decomposition of the diagonal correspondence (not
a numerical equivalence). This is the smooth-core analog of the
surface corrections from Part 5.
Proposition 6.4 (Controlled pure lifts). For every stable heavy
type (h,B) one can choose a symmetric class G_{h,B} in
A^2(X_{h,B}(3)) with the following properties:
(i) Over the smooth heavy base, G_{h,B} equals the pullback of
Delta_C^pure by the coarse-position map.
(ii) Pull a raw cubic F^0 or one of these chosen pure lifts
G_{h,B} from any ancestor chart to a compact descendant
cut chart. Its operation is a sum of the chosen pure
lifts at relevant descendant vertices (all three labels
at that vertex), together with pair-times-singleton
operations and a residual cubic operation.
(iii) The singleton operations in (ii), and all their further
pullbacks through cuts and light forgettings, lie
componentwise in the span of unit, canonical class, and
flag points (over fixed smooth core data). On any smooth
test family the same holds modulo classes supported
over positive base codimension.
(iv) On the current smooth-core open, the residual operation
in (ii) is a finite sum of classes containing a pulled
base divisor factor (from current or ancestor heavy base).
Consequently its pullback to every smooth test family is
zero operationally after removing a positive-codimension
subset of that test base.
All expansions preserve the color multiset. They may always use
one and the same chosen G and centering at a given current
vertex, regardless of which ancestor supplied the operation.
Proof sketch. On the smooth heavy base of the class itself, the
blowup decomposition (**) and Lemma 6.3 express F^0 as the
pullback of the small diagonal, plus pair-exceptional-times-
singleton terms and triple-exceptional-times-base-divisor terms.
Subtract these, and also subtract the [Delta] - Delta^pure
decomposition.
These subtractions have compact extensions with the required
structure:
- pair exceptional extends as the heavy-flag tail boundary on
the two-light space;
- triple exceptional extends as its three-light counterpart;
- divisor coefficients extend by Chow localization;
- singleton factors use units, heavy sections, relative
dualizing line on one-light universal curve, and base
classes.
Define G by making these structured subtractions from F^0 and
averaging over the three labels. This gives (i).
For (ii), pull to a deeper cut chart. The raw F^0 has the
decomposition just proved there. Its main component is a sum of
small diagonals of relevant descendant vertices. Replace each by
the corresponding G plus structured corrections.
For (iii): inherited pair-times-singleton subtractions retain
the form; the singleton spans as claimed by the normalization
formula for the dualizing line (canonical class plus branch
points on each component, base terms stay base terms).
For (iv): the only other corrections are exceptional divisors
times base divisor classes. On pulling an inherited correction,
the exceptional factor need not have a special form, but its
base divisor factor remains pulled from an ancestor heavy base.
So the residual is of the form
sum_j H_j * rho_j^* D_j,
H_j a one-codimensional three-label operation,
D_j a divisor class on a current or ancestor heavy base.
Represent each pulled rational divisor class by the first Chern
class of a rational line bundle. After removing the finite union
of divisors on the test base, every summand is zero as an
operational class. This remains true when S maps into a special
locus of an ancestor base: what matters is the pulled line
bundle on S, not the original support of D_j.
Assignment conditions can be imposed separately on pair and
singleton factors by making their operations zero on other
components of the disjoint union X^Gamma(k). All steps occur
before inserting colors, so preserve their multiset.
End of Proposition 6.4 sketch.
--------------------------------------------------------------------------------
6.5 Support advancement
--------------------------------------------------------------------------------
We now formalize the induction that repeatedly lowers the
dimension of the base image of a residual cycle.
Setup. A "term" in the induction is a set of active operations,
pulled to disjoint sets of light labels, acting on a cycle on
the full labeled chart. The cycle need not factor over the
labels or the vertices. This is the freedom that lets us push
through the induction even after odd operations have "consumed"
some labels.
Lemma 6.5 (Support advancement). Let a cycle multiplier on
X^Gamma(n) have base image contained in an irreducible closed
Z subset of B_Gamma. After a refinement if necessary, suppose
its generic base image lies in the smooth core S_Gamma. Let L be
a factor involving a specified subset of the active entries.
Suppose that the pull of L vanishes operationally over a dense
open of a smooth test family dominating Z intersect S_Gamma.
Then its action can be written as a finite sum of cycles with
strictly smaller base-image dimension. Only the entries in L
are consumed.
More precisely: over a smooth test base S proper generically-
finite over Z intersect S_Gamma, if the vanishing holds off
base codimension >= a, then the terms remaining over S_Gamma
have base-image dimension at most dim Z - a. Other terms lie
over Z minus S_Gamma and require a strict graph refinement.
If one replaces S by the total space of one of its curve
families (to get a section), the corresponding bound is
dim Z - a + 1.
Proof sketch. A cycle with a finer generic dual graph can be
lifted to a chart of that graph; heavy gluing is finite, so
this doesn't increase base-image dimension.
Use the following property of rational Chow groups: if
T' -> T is proper and surjective, every integral cycle on T has,
after multiplication by a nonzero rational, a proper lift
generically finite over it. Close up a closed fiber point, and
the pushforward is the original cycle times the generic degree.
Same property holds for finite-type Deligne-Mumford charts
here [BaeSchmitt B.19].
For the test base, take a finite level cover of the smooth
heavy-curve stacks and resolve the component over
Z intersect S_Gamma (Wlodarczyk). A full level of order >= 3
removes automorphisms in positive genus (Deligne-Mumford,
Deligne-Rapoport). Lift the cycle multiplier properly.
Let A subset S be codim >= a, with L operationally zero off A.
Chow localization expresses the action as a pushforward of
cycles lying over A. Return to the original family by
projection/pushforward. These cycles have base-image dimension
< dim Z. Closing up inside the preimage of Z introduces cycles
only over Z minus S_Gamma, a proper closed subset.
To get a section of a smooth curve family C -> S, pull back
along C -> S, where it has its diagonal section. New test base
is smooth of dim Z+1. A support of codim >= a in the new base
has image dim <= dim Z - a + 1.
All operations concern the chosen factor L. Other operations are
carried along by pullback and projection formula.
End of Lemma 6.5 sketch.
--------------------------------------------------------------------------------
6.6 Elementary vanishing tests and repeated color
--------------------------------------------------------------------------------
We need concrete tests that provide the operational vanishing
required by Lemma 6.5.
Define, for a graph Gamma,
R_Gamma = 2|V(Gamma)| + 2|E(Gamma)|.
Lemma 6.6 (Noise and small-genus tests). If R_Gamma + 1 noise
entries have their singleton at the same color, their product
advances support while consuming those entries. If a genus-h
vertex has more than C(2h,3) pure entries, the product of any
C(2h,3)+1 of them advances support.
Proof sketch. Over smooth generic core data, the one-light space
at a vertex is its curve. Permitted singleton classes lie in
the span of unit, canonical class, and flag points, so overall
dimension is at most
sum_v (2 + |B_v|) = 2|V| + 2|E| = R_Gamma.
A one-slot operation is odd in the sign-coinvariant algebra, so
a product of R_Gamma + 1 same-color singletons vanishes by
alternation. Accompanying pair operations can be grouped
separately.
For the pure entry bound: by Proposition 5.1, each pure entry
contributes a value of the same odd cubic phi in
B_h (x) Lambda^3 W_h^dual. Values span at most C(2h,3) odd
coefficients; a longer product vanishes by supercommutativity.
Faithfulness gives the operation vanishing modulo positive base
support. Combine with Lemma 6.5.
End of Lemma 6.6 sketch.
Lemma 6.7 (Repeated-color advancement). Suppose a vertex of
genus h has a flag and k > C(h+1,2) pure entries contain one
common color i. Then their product is a finite sum of products
L * H with:
(i) L involves at most h+2 entries and vanishes modulo
positive base support;
(ii) H is still a product of individual pure entries at that
vertex;
(iii) each term has the same color multiset as the original.
Consequently each term admits a support advancement consuming at
most h+2 entries.
Proof sketch. A symmetric cubic class defines an alternating
three-color operation. Put the common color first in each entry;
the product then factors through Lambda^k(Lambda^2 U_0). This
holds already on the compact chart (alternation inside a pair is
from cubic symmetry; interchange of two cubic entries has sign
minus because arity is odd).
By Schur-Weyl and the row cutoff, the corresponding projector p
on the 2k input slots lies in the two-sided ideal of the group
algebra generated by the alternator on h+2 slots. In a finite
sum decomposition, each term uses at most h+2 entries during
alternation. Group those into L; the remaining entries go into H.
For vanishing of L: work modulo positive base support, center
the curve projector at the flag section. Then int(e) = 1,
int(e^2) = 0, evaluation at the section is multiplicative and
kills W_h. For standard odd vectors u,v,
u v = omega(u,v) e + (u v)_pure
(no unit component, since coefficient of u v in 1 is its
evaluation at the section).
Test via z: U_0 -> W_h, a = z(i). Each entry becomes
T_a(u,v) = phi(a,u,v), an alternating 2-form with odd
coefficients. The critical fact: contracting a variable slot
between two such entries gives zero (super: a odd, a^2 = 0 in the
Frobenius contraction int(a u * a v)).
After alternation, L is an ordinary alternating (h+2)-form on
W_h. Every summand of its double symplectic contraction is
zero: if a contraction joins distinct entries, do it first and
get zero; otherwise two internal contractions in distinct
entries give (odd)^2 = 0.
For h >= 2, double contraction
Lambda^{h+2} W_h^dual -> Lambda^{h-2} W_h^dual is injective by
symplectic Lefschetz. For h=0,1, source alternation is already
zero. All color tests of L vanish. Apply Proposition 5.1 and
Lemma 6.5.
End of Lemma 6.7 sketch.
--------------------------------------------------------------------------------
6.7 Finite color budget
--------------------------------------------------------------------------------
Fix positive INTEGER constants
K = 10^5,
m_0 = 2000 * g^{2/3},
C_K = C(2K, 3).
Reserve 4K + 20 disjoint blocks of m_0 colors each. Let S_0 be
the remaining colors. We use
(4K + 20) m_0 / g = 8.0004 * 10^-12 < 0.1,
|S_0| > 0.9 g.
For a current graph put
b = max_v h_v, l = g - b.
Under refinement the maximum genus cannot increase (so l cannot
decrease), and the number of edges cannot decrease either.
Unmarked stability gives
|V| <= 2g - 2, |E| <= 3g - 3, R_Gamma <= 10 g.
Advancements fall into two regimes:
Ordinary advancements: cost entries, from residual, noise,
small-genus, or repeated-color tests. Each consumes at most
~11g entries.
Expensive advancements: from Lemma 5.6 (long products of pure
entries), only when a single vertex carries almost all genus.
These lower base-image dimension by >= g - K - 2.
We count available entries against these costs. The key
inequality is
10^17 g << 0.01 D_0 for D_0 = C(m_0, 3),
which holds easily at g = 10^60. The reserve blocks suffice for
the total of at most 3g - 2 advancements and at most
3K + 5 expensive ones. We do not need the details here; the
bookkeeping lives in Section 8 of the original.
--------------------------------------------------------------------------------
6.8 Completion
--------------------------------------------------------------------------------
Theorem 6.8. The class Y of Definition 3.4 satisfies
q_{g,n}(Y) = 0 in A^*(Mbar_{g,n}; Q).
Proof sketch. Begin on X_{g,empty}(n) with the product of cubic
operations F^0_{g,empty}, one for each increasing color triple,
and the fundamental cycle as multiplier. Apply Proposition 6.4
and the preceding induction (Lemmas 6.5, 6.6, 6.7).
At every stage, each nonzero branch admits an advancement: use a
residual or noise test when available, the small-vertex and
reserved-block tests if l < K, and otherwise the repeated-color
test. The counts show all required active entries and unused
blocks are available at the moment they are needed.
Each individual expansion and localization involves finitely many
cycles. Each branch has length at most 3g - 2 (the advancement
bound from Section 8 of the original paper). The process therefore
ends with empty support, proving that the full product is zero in
the sign-coinvariant Chow group of the weighted space.
The weighted-to-unweighted comparison (Lemma 6.1) identifies the
pullback of the raw cubic with q(F_J^st). After signed averaging,
the coefficient of the ordered color word i is a nonzero rational
scalar times q(Y). Pullback of the zero weighted class proves
the theorem.
End of Part 6.
=====================================================
PART 7: WHAT THIS CHANGES
=====================================================
We've now seen the whole proof. Here: what it means in
context, and what remains open.
--------------------------------------------------------------------------------
7.1 The main consequence
--------------------------------------------------------------------------------
Pixton's original 2012 conjecture asked whether his proposed
system of stable-graph relations was COMPLETE: whether
ker q_{g,n} = P_{g,n} for every stable (g,n). This paper shows
the answer is no: there is at least one stable pair for which
ker q is strictly bigger than P.
The proof is a single counterexample. That's enough. Once
completeness fails somewhere, the uniform conjecture is false.
Equivalently: the family of Pixton relations, though it produces
many true relations, is not exhausted by the ones the recipe
gives. There is a tautological relation of a different kind,
one that no amount of Pixton-recipes (multiplied, glued,
forgotten) can produce.
The specific witness Y is amusingly small in a certain sense:
it is a codimension-2 D_* class, built from pairs of rational
tail divisors and a rational-bridge correction, alternated
against color-symmetric shuffles. Its existence was not
suspected from any particular low-complexity example, which is
why the conjecture lasted as long as it did.
--------------------------------------------------------------------------------
7.2 Pixton's own follow-up work: non-Gorenstein at M_25
--------------------------------------------------------------------------------
Completeness is a different question from the GORENSTEIN
PROPERTY.
- The Gorenstein property asks: is multiplication into
complementary degree a nondegenerate pairing?
- Completeness asks: are all relations among generators
produced by Pixton's rules?
These are independent questions. The Gorenstein property can
fail while completeness holds, and vice versa.
Pixton himself proved that R^*(M_25) is not Gorenstein
(Pixton 2026, Theorem 1.1, Section 2.2), while establishing
Faber-Zagier completeness in that genus. That is a case where
the "nice geometric property" fails but the "list of relations
is complete" property holds.
So the two failure modes are truly different, and this paper's
failure is of the "completeness" kind. That's important
conceptually: you cannot use this paper to deduce anything new
about Gorenstein-ness at M_25, or vice versa.
--------------------------------------------------------------------------------
7.3 Related work
--------------------------------------------------------------------------------
Canning-Larson-Schmitt (2026). Proves completeness together with
non-Gorenstein behavior on the COMPACT-TYPE spaces
M^ct_6, M^ct_{5,2}, M^ct_7. These are compactifications that
include only curves of compact type (no separating nodes). The
technique relies on the special structure of compact-type. Our
counterexample lives on the FULL stable compactification; there,
Pixton's rules are not complete. So there is no contradiction:
completeness on a smaller space (compact type) does not imply
completeness on the full Mbar.
Janda's discriminant paper (2018). Compares two methods for
producing relations from reconstructions of semisimple CohFTs
with nonempty discriminant. That comparison identifies the
systems arising from two different cancellation schemes.
It does NOT identify the entire Chow kernel with Pixton's
system. Our paper's comparison is with the specific Pixton
system P_{g,n}; Janda's comparison is orthogonal to that.
PPZ (2015) and Janda (2017). Proved validity of Pixton's
system: P_{g,n} is a subset of ker q_{g,n}. These proofs use
3-spin CohFT and equivariant GW theory of P^1 respectively.
Validity is a strong form of "these are indeed relations".
Completeness is the reverse direction, and this paper refutes it.
Canning (2025). Surveys the tautological ring and its
Gorenstein property for full Mbar and various subspaces,
discussing exactly the "original system" being refuted here.
That paper states the Chow and rational-cohomological
completeness of the original Pixton system as a conjecture
(Conjecture 1 in Canning 2025). This paper provides a
counterexample to that specific statement.
--------------------------------------------------------------------------------
7.4 Different flavors of "Pixton conjecture"
--------------------------------------------------------------------------------
One caution about language. The phrase "Pixton's conjecture" is
sometimes used to refer to different statements depending on
the community. Three distinct versions:
V1. Original Pixton 2012 graph formula gives a COMPLETE system.
V2. Weaker version restricted to specific colors, specific
decorations, or specific subspace of Mbar_{g,n}. Failure
of V1 does NOT refute all such weak versions.
V3. Complete-system conjecture only for the "operational"
closure P_{g,n} (Pixton span including all Pixton-ideal
consequences).
This paper refutes V1 and V3, but does not directly refute
narrower V2 variants that, e.g., restrict to compact type
(Canning-Larson-Schmitt have positive results there) or to
low genus (Pixton at M_25).
When you see "Pixton's conjecture is false" in the popular
press: check which version is being discussed. This paper
specifically refutes the ORIGINAL full-system version, and
that's what makes it non-trivial.
--------------------------------------------------------------------------------
7.5 What remains open
--------------------------------------------------------------------------------
Three directions remain open and interesting:
(a) Other counterexamples in lower genus. The construction
uses g = 10^60 to make the color budget and the
support-dimension clear. There may be infinitely many
lower-genus counterexamples, and characterizing the
exact set of (g,n) where Pixton completeness holds is
open. This paper only refutes the conjecture at one
explicit (g,n).
(b) Characterize ker q. This paper's Y is a specific class
that's in ker q but not in P. What the FULL kernel is
(i.e., how to describe an enlarged relation system that
IS complete) remains open. What kind of relations do you
need to add to Pixton's system to get ker q exactly?
(c) Gorenstein-ness beyond M_25. Pixton proved
non-Gorenstein at M_25. Canning-Larson-Schmitt
completed compact-type cases. The full
R^*(Mbar_{g,n}) Gorenstein behavior across all (g,n)
remains under investigation. This paper doesn't touch
that question except by example: the failure of
completeness does NOT imply failure of Gorenstein.
Beyond these three, there is a longer-term question: what is
the right conceptual framework for the tautological ring that
would give completeness in a broader sense? Candidates exist
(e.g., adding cross-boundary "Mumford-type" relations, or
working in a quotient that naturally produces the correct
kernel). This paper's Y is meant to be a stepping stone:
it tells you the original system was too small, and the proof
techniques (exterior-ring detectors, surface Quot relations,
support induction) may generalize to a full description.
--------------------------------------------------------------------------------
7.6 Summary of what just happened
--------------------------------------------------------------------------------
To recap, in one paragraph: we built an explicit class Y in
the strata algebra S_{g,n} for g = 10^60 and n = 3*C(g,3). We
proved two independent facts about Y:
(i) Y is not in the Pixton span P_{g,n}. (Part 4: formal
detector over an odd Frobenius algebra.)
(ii) Y vanishes in rational Chow. (Parts 5-6: surface
Quot relation plus compactification plus support
induction.)
Together, (i) and (ii) say ker q_{g,n} is strictly bigger than
P_{g,n}. By the cycle-class map, the same failure holds in
rational cohomology. This refutes Pixton's original
completeness conjecture.
The two proofs live in completely different worlds (formal vs
geometric) and neither implies the other. This is why the whole
apparatus -- surface Quot schemes, Hilbert-scheme scalars,
support induction -- is necessary. If we could prove non-membership
from geometry alone, or vanishing from formalism alone, we
wouldn't need either the detector or the surface machinery.
That's the paper. The glossary follows.
=====================================================
GLOSSARY
=====================================================
The alphabetized terms appear throughout. Entries that are
fairly technical (marked with *) are only needed if you want
to read the formal proofs.
A^* (X), Chow group of X
Algebraic analog of cohomology. Cycles modulo rational
equivalence, graded by codimension. Fact: A^*(X) is a
graded commutative ring.
A-curve
Algebraic curve.
A compact smooth complex curve of genus g. So it is
simultaneously a Riemann surface of genus g and an
algebraic curve over C.
algebraic curve of genus g
A compact smooth complex curve (Riemann surface) with
g "holes".
Aut Gamma (automorphism group of a graph Gamma)
Permutations of half-edges preserving the graph structure
and the marked vertices. Finite. Same for stable graphs.
bar transform (*)
The change of variables psi -> psi_bar and kappa -> kappa_bar
that separates boundary divisors from the "root" part.
bar_psi, bar_kappa (*)
psi and kappa after subtracting all boundary contributions.
The formal "root" cotangent and kappa.
B_b (faithful Chow coefficient algebra) (*)
A supercommutative rational algebra constructed in Part 5
so that the tensor model of curve correspondences is
faithful. Contains all Chow correspondences between curve
powers, not just pairing diagrams.
b (in the surface Quot section)
The genus of the smooth curve family underlying the
surface. The "b-1" appears in base codimension bounds
because the surface Quot relation holds modulo that.
base-support quotient (*)
The quotient of A^* by classes vanishing off a closed
subset of codimension at least a. Used to track loss
of information on parameter families. Not numerical
equivalence.
beta_J^st, beta_{h,B} (bridge class)
A codimension-two class summing over one-edge unmarked
graphs of the relevant type, subdivided by a
genus-zero vertex carrying the three labels.
Chow group (see A^*)
Algebraic cycles modulo rational equivalence.
CohFT
Cohomological Field Theory. An algebraic structure encoding
tautological relations from Gromov-Witten theory. Not
needed for our proof, but historically crucial for
Pixton validity.
compact type
A boundary stratum with only compact-type stable curves
(no separating nodes). M^ct_{g,n} is open inside Mbar_{g,n}.
coproduct (*)
The adjoint of multiplication with respect to the
Frobenius pairing. Delta_L: V -> V^{(x)L} for a label set L.
cycle class map cl
A^* (X) -> H^{2 *}(X), the map from Chow groups to
singular cohomology (over Q). Used for Corollary 2.2.
D_T (tail divisor)
A codimension-one boundary divisor on Mbar_{g,n}
consisting of curves where the labels in T sit on a
rational tail attached to the rest by a single node.
D_bar_{ab}
Sum over all T containing {a, b} of D_T.
Delta_C, Delta_C^pure (*)
Small diagonal in C x_S C x_S C, and its projection via
eta_C in each of the three curve slots. Delta_C^pure is
a pure odd-class in the tensor model.
Delta_e (edge series) (*)
Pixton's edge function in T, built from A(T), B(T) with
the identity A(T)B(-T)+A(-T)B(T)+2=0.
e'_C (*)
e_C - (1/2) pi^*(a_C). A corrected degree-one divisor
class. Ensures the "hyperbolic plane" decomposition works.
eta (*)
[Delta_C] - e'_{C,1} - e'_{C,2}. A projector onto the
odd symplectic summand.
eta_C (*)
Same as eta but for a smooth core curve pi: C -> S.
exterior algebra
Lambda_Q(...) means: multiply symbols antisymmetrically.
Square of any odd generator is zero.
F^0_{h,B} (corrected cubic)
The raw boundary cubic on the weighted compactification:
[1=2=3] - (1/2) beta_{h,B} + (1/2) sum_p delta_{p,123} psi_p.
F_J^st (three-slot factor)
D_bar_{ab} * D_bar_{ac} - (1/2) beta_J^st, the building
block of Y.
Faber-Zagier
An early system of relations among kappa classes on
M_g (smooth curves). Its proving (Pandharipande-Pixton
2021) motivated Pixton's extension to Mbar.
Frobenius algebra (*)
An algebra V with a nondegenerate pairing (the integral),
such that V is self-dual for that pairing. The odd
symplectic examples here carry both a symplectic form
and an alternating cubic form.
g, n (stable pair parameters)
g = genus, n = number of marked points. Stability is
2g-2+n > 0. The paper's counterexample uses g = 10^60
and n = 3*C(g,3).
Gorenstein property
The tautological ring R^*(Mbar_{g,n}) is Gorenstein if
multiplication R^d x R^{top-d} -> R^{top} is a perfect
pairing. This is a "nice geometric property". It is
independent from Pixton completeness.
Gross-Schoen cycle (*)
A corrected small diagonal in the triple product of a
pointed curve. The prototype of the projected small
diagonal in Part 5.
Hassett (weighted) moduli space
Mbar_{g,(w_1,...,w_n)} for positive rational weights
summing to at most 1. Allows light weights to collide.
Named after Brendan Hassett.
heavy / light marks
In weighted stable curves, "heavy" marks (weight 1) are
kept separate; "light" marks (weight epsilon) may collide.
Hilbert scheme of points (X^{[r]})
Moduli of finite flat subschemes of length r. Smooth if
X is a smooth surface. Used in the surface Quot relation.
kappa class
A class measuring how the marked points wrap around a
curve. kappa_j = pi_*(psi^{j+1}) where pi forgets an
extra mark. In the strata algebra: kappa_j is a formal
symbol at each vertex.
ker q_{g,n} (the kernel)
The set of formal combinations in S_{g,n} mapping to zero
under q_{g,n}. These are the actual relations.
lambda
(2g-2) e in the odd Frobenius algebra. Plays the role of
the "handle element" at a curve. Properties: lambda^2 = 0.
Losev-Manim space
Mbar_{0,(1,1,epsilon,epsilon,epsilon)}. The weighted
chain space with two heavy endpoints and three lights.
In Lemma 6.3, it is the first blowup P^2 of the
triple-exceptional locus.
M_{g,n}, Mbar_{g,n}
Moduli of smooth curves / stable curves of genus g with
n marked points.
marked point
A distinguished labeled location on a curve.
node
A transverse self-intersection of a curve.
Pixton relation R(g,n,r;sigma,a)
The formal sum over stable graphs of [Gamma, P_Gamma] /
|Aut Gamma| built from the recipe of Section 1.7.
Pixton span P_{g,n}
The formal span of all Pixton relations, multiplied and
glued by arbitrary auxiliary strata. Already an ideal
(Pixton Prop 1).
psi class
The cotangent class at a marked point (or at a
half-edge). psi_i in A^1(Mbar_{g,n}).
q_{g,n} (realization map)
S_{g,n} -> A^*(Mbar_{g,n}), the formal-to-geometric map
that sends [Gamma, gamma] to xi_{Gamma,*} gamma.
Unnormalized (no Aut denominators).
Quot scheme
Moduli of quotients of a fixed coherent sheaf. In our
surface setting, parametrizes rank-zero quotients of
V = K + O_X with fixed determinant and ch_2.
rational tails graph (*)
A tree with one genus-g root and all other vertices
rational, every rational branch having >= 2 descendant
labels. Not the same as "curves with rational tails".
rational tail divisor
Same as a tail divisor D_T (see above).
S_{g,n} (strata algebra)
The free Q-vector space on decorated stable graphs modulo
isomorphism, with per-vertex dimension truncation.
small diagonal
The diagonal Delta in C x C x C (three copies). Its
"projection" via eta corresponds to a modified diagonal
of Gross-Schoen type.
stable graph
A graph with genus-labelled vertices whose dual graph
is the combinatorial shadow of a stable curve.
Stable pair (g,n)
2g - 2 + n > 0. Necessary and sufficient for the moduli
space Mbar_{g,n} to exist as a nonempty stack.
support induction
The method of reducing the codimension-a support of a
cycle until it vanishes, used in Part 6.
symplectic space (*)
A vector space with a nondegenerate antisymmetric form
< , >. In the odd setting, the form provides the pairing
on V's middle summand.
tautological (class / ring / space)
Classes built from psi and kappa and gluing. The full
subring R^* = q(S) of A^*(Mbar_{g,n}) is the
tautological ring.
Theta = product_{i<j<k} theta_{ijk}
The top exterior monomial. Nonzero in R. This is the
value that t_n(Y) reduces to, up to a nonzero scalar.
theta_{ijk}
Odd generator in the exterior ring R for each increasing
triple of colors. Alternating in i,j,k.
t_0 = 25 chi - 1 (chi = (b-1)^2)
The power appearing in the surface bound: T_gamma^{t_0}=0.
t_m (formal detector) (*)
The recursive Q-linear map S_{g,m} -> V^{xm} that kills
the Pixton span but satisfies t_n(Y) != 0.
X_{h,B}(k) (weighted space)
Hassett space with k light weights added to heavy data (h,B).
Y (the counterexample class)
Definition 3.4. Sum over signed permutations of the color
word of product_J F_J^st. Lives in S_{g,n}^{2D_*}.
xi_Gamma (gluing map)
product_{v} Mbar_{g_v,n_v} -> Mbar_{g,n}. The map defining
q_{g,n}.
zeta_v (parity variable) (*)
Zeta_v^2 = 1 at a vertex v. Controls the parity of kappa
classes in Pixton's cycle formula.
=====================================================
END
=====================================================
The mathematics of the actual paper is much denser, but the
ideas you just absorbed -- formal vs geometric, kernel vs
generators, support induction, faithful tensor tests -- are
exactly the tools mathematicians use at the research edge.
The rest is filling in conventions.
If you want to go further, useful starting points:
- The moduli space of curves (Harris-Morrison,
"Moduli of Curves"). For the geometric picture.
- Intersection theory (Fulton, "Intersection Theory").
For the algebraic machinery.
- The Pandharipande-Pixton-Zvonkine papers
(arXiv:1303.1043) for 3-spin relations.
- Pixton's original (arXiv:1207.1918) for the
relation recipe.
- This paper's original LaTeX sources for the
formal details.
End of the rewrite.