A high-arity counterexample to Pixton completeness in Chow

1 Introduction

The tautological ring of the moduli space of stable curves is generated by cotangent and kappa classes, together with the classes obtained by forgetting markings and gluing nodes. Describing all relations among these natural classes is a central problem in the intersection theory of moduli spaces. Pixton proposed a uniform system of stable-graph relations and conjectured that it is complete. We construct a relation outside that system: an alternating product of corrected small diagonals that vanishes in rational Chow, and therefore in rational cohomology.

1.1 The relation problem

Mumford initiated the systematic study of natural classes in the Chow ring of the moduli space of curves [Mum, Sections 4–5]. Faber’s calculations and geometric constructions on powers of the universal curve led to a proposed description of the tautological ring of the smooth unpointed space \mathcal M_g, including its Gorenstein property [Fab, Conjectures 1–2]. Faber and Zagier later gave explicit hypergeometric formulas for relations among the kappa classes. Pandharipande–Pixton proved these relations in Chow using stable quotients and formulated their completeness on \mathcal M_g [PP, Theorem 1 and Conjecture 2].

On the compactification, a presentation must also account for boundary strata and their intersections. Arbarello–Cornalba and Graber–Pandharipande developed the forgetful and boundary calculus [AC, Section 1], [GP03, Appendix A]; in genus zero, Keel gave a presentation of the entire Chow ring by boundary divisors [Keel, Section 4, Theorem 1]. Pixton’s 2012 graph formula extends the Faber–Zagier system to \Mbar_{g,n} and proposes a complete set of relations [Pix, Sections 1–3, Conjecture 2]. Pandharipande–Pixton–Zvonkine proved the cohomological validity of this system using the 3-spin CohFT [PPZ]; Janda proved its Chow validity using equivariant Gromov–Witten theory of \mathbb P^1 [Jan]. These results establish relations, while completeness asks whether any further relations exist.

We fix the original relation space precisely. Let \Salg_{g,n} be the strata algebra of formal decorated stable graphs, with decoration degree truncated at the dimension of each vertex factor. Its realization is

The gluing pushforward generators are unnormalized, as in [PPZ, Section 0.3]. Let \Pspan_{g,n} be the span obtained by placing one of Pixton’s graph-formula relations at a vertex of an outer stable graph, placing arbitrary strata classes at the other vertices, and gluing. Pixton’s Proposition 1 shows that this span is already an ideal, stable under the gluing and forgetting pushforwards and pullbacks [Pix]. Thus it includes these formal consequences from the outset. The smaller 0/1-marking seed system recovers the same span under the operations described in [PPZ, Section 3.5]. Section 2 gives the full recipe and all conventions.

Writing R^*(\Mbar_{g,n})=\im q_{g,n}, the Chow completeness question is whether

for every stable pair. The known validity theorem gives the inclusion \Pspan_{g,n}\subseteq\ker q_{g,n}. Our example makes it strict.

1.2 The counterexample

The construction uses g colors and one three-label factor for each increasing triple of colors. Each factor is a product of two rational-tail divisor sums, corrected by a class supported on two-node rational bridges. We multiply these factors and alternate over relabelings that preserve the color of each slot. Definition 2.3 specifies the resulting formal class Y.

The same class gives the rational-cohomological consequence. To specify that formulation, write q^{\mathrm{CH}}_{g,n}=q_{g,n} and define, in formal degree d,

where \operatorname{cl} is the rational cycle-class map for the complex Deligne–Mumford stack. Its image is the cohomological tautological ring RH^*=\operatorname{cl}(R^*). Completeness in both R^* and RH^* is explicitly recorded as Pixton’s conjecture in [Can, Section 1, Conjecture 1].

The theorem and corollary give negative answers to the Chow and rational-cohomological completeness questions for the full original system. The large parameters leave ample room in the support and color estimates; they are not proposed as minimal parameters.

Completeness is distinct from the Gorenstein property. The latter asks for nondegenerate multiplication pairings in complementary degrees, whereas completeness asks for all relations among the generators. Pixton proved that R^*(\mathcal M_{25}) is not Gorenstein while establishing Faber–Zagier completeness there [Pix26, Theorem 1.1 and Section 2.2]. Canning–Larson–Schmitt proved completeness together with non-Gorenstein behavior on the compact-type spaces \mathcal M^{\mathrm{ct}}_6, \mathcal M^{\mathrm{ct}}_{5,2}, and \mathcal M^{\mathrm{ct}}_7 [CLS, Theorems 1 and 7]. Our relation lies on the full stable compactification.

There is also a distinction between completeness and comparison of methods for producing relations. Janda’s comparison theorem identifies the systems arising from cancellation of reconstruction singularities for convergent semisimple CohFTs with nonempty discriminant [JanD, Definition 3.3.1 and Theorem 3.3.6]. It concerns the relations generated by those reconstructions; it does not identify the entire Chow relation kernel. The comparison throughout this paper is with the explicitly fixed span \Pspan_{g,n}.

1.3 Proof and methods

The proof compares two algebras of cubic coefficients. A formal super Frobenius algebra can carry independent odd cubic coefficients while satisfying the identities needed to annihilate the original Pixton span. The actual Chow correspondences of a curve satisfy an additional nilpotence identity obtained from a surface Quot scheme. The first algebra proves nonmembership; the second proves vanishing. Figure 1 separates these two assertions about Y.

The formal test. Section 3 constructs a linear map on decorated graphs by discarding graphs outside the rational-tails locus and evaluating rational branches by Frobenius multiplication and coproduct. This uses the graph calculus familiar from two-dimensional topological field theory [Abr, Section 2 and Theorem 3]. The odd symplectic curve pairing and the vanishing of contractions between isotropically supported corrections have precedents in Pandharipande–Zvonkine’s CohFT constructions [PZ, Sections 1.1–1.5 and Remark 11]. Here the coefficient ring is the exterior algebra on independent \theta_{ijk}, one for every color triple; we verify the graph identities directly. Every element of \Pspan_{g,n} has zero test, while the test of Y has a nonzero coefficient proportional to \prod_{i<j<k}\theta_{ijk}.

The smooth-family Chow identity. Section 4 splits the correspondence object of a smooth curve into two even summands and an odd symplectic summand. The coefficient-algebra construction is a concrete form of O’Sullivan’s semisimple tensor reconstruction [OS, Lemma 2.3.1]. It retains all actual relative Chow correspondences, so its faithfulness is an assertion about Chow, not about cohomological realization. The small diagonal projected to three odd summands gives an alternating cubic \varphi with odd coefficients. Over a pointed curve this is the modified diagonal of Gross–Schoen [GS, Equation (0.1) and Sections 2–3]; the family construction also retains the necessary base corrections.

Sections 5–6 prove the additional nilpotence. The surface is the square of the curve. A relative Quot scheme with fixed determinant produces a relation for products of the cubic on isotropic vector sets. Its obstruction theory uses derived perfect-complex moduli and determinant maps [TV, STV], virtual pullbacks [BF, Man], and virtual localization [GP]. Surface Quot localization and relative Chow identities appear in Marian–Oprea–Pandharipande [MOP1]; the point obstruction factors have the antecedent in [OP, Lemma 34 and Section 5.2]. Here the calculation keeps the determinant constraint, arbitrary irregularity, and every marked surface output.

The nested-Hilbert recursion of Ellingsrud–Göttsche–Lehn [EGL] reduces that marked calculation to decorated equality diagonals. Kong develops relative forms of this recursion and, for curve families, formulas retaining one and two marked outputs [Kong, Section 2.1 and Lemma 3.4]. We prove the version retaining an arbitrary finite set of surface outputs. Only bare triple blocks survive the relevant codimension bound. Section 6 computes their common scalar and proves it is nonzero, using symmetric-power and residue methods related to [MOP2, Equation (9), Theorem 2, and Lemma 3]. It then converts the isotropic identity into vanishing tests for sufficiently long products with arbitrary color inputs (Lemma 6.4). Finite faithful polarization returns a Chow identity modulo classes supported over the prescribed positive codimension in the base.

Passage to the compactification. Section 7 compares the corrected stable cubic with these projected smooth-curve cubics. The framework uses Hassett’s weighted stable curves [Has] and configuration-space blowups [FM, Rou]; Petersen’s proof of Tavakol’s presentation conjecture treats the corresponding rational-tails comparison over a smooth core [Tav, PetW]. The nodal comparison here requires a rational-bridge correction with coefficient -1/2. A three-position calculation shows that it cancels the remaining triple-exceptional contribution. The errors are then either a two-slot operation times a one-slot operation from a bounded span, or operations supported over positive base codimension.

Section 8 uses these errors to lower the dimension of the image of the supporting cycle in the base. Chow-localization support methods occur in Graber–Vakil’s filtration by rational components [GV, Theorem 1.1 and Section 4.2]; the induction here acts on arbitrary cycle multipliers over the base parametrizing the curve components. When one component carries almost all the genus, the surface identity advances support. Otherwise a repeated-color identity supplies a smaller vanishing factor, using the curve algebra centered at a marked node branch, Schur–Weyl alternation, and symplectic contraction. The final color budget leaves enough entries for every advancement, until the support is empty. Pullback from the weighted space then proves q_{g,n}(Y)=0.

Thus Sections 2–3 construct and detect the formal candidate; Sections 4–6 prove the smooth-family identity; and Sections 7–8 extend its vanishing to the compactification. Table 1 collects the notation used across these parts.

1.4 Conventions

All varieties, stacks, and families are over \C, and Chow groups have rational coefficients. On a smooth space we identify Chow classes with operational classes when pulling them to a test space. Products of stacks are understood in rational Chow theory [Kre, Theorem 2.1.12]. We use proper rational pushforward for relative Deligne–Mumford maps and its compatibilities from [BSch, Theorem B.17 and Propositions B.18–B.19]. Smooth Deligne–Mumford stacks have the operational identification in [BSch, Theorem C.6 and Corollary C.9]; for smooth quotient stacks, see also [EG, Propositions 4 and 19]. Every formal series calculation is truncated at a specified coefficient, so only finitely many terms occur.

Parity is distinct from Chow codimension. In a super tensor product, interchanging homogeneous factors of parities p,q gives (-1)^{pq}. In particular, on an odd vector space the super-symmetric tensors are ordinary alternating tensors. All finite averaging and polarization arguments take place in characteristic zero.

2 The strata algebra and the explicit class

This section fixes the relation space. The distinction between an unnormalized gluing generator and an automorphism-weighted graph sum will be used in both parts of the proof.

2.1 Decorated graphs

A stable graph \Gamma of type (g,n) is a connected graph with n labeled legs and genera g_v\geq0 at its vertices, satisfying

Here n_v counts all incident half-edges, including legs. Its gluing map is

At every half-edge let \psi denote the cotangent class. At a vertex define

where \pi forgets an additional marking.

The vector space \Salg_{g,n} is spanned by formal symbols [\Gamma,\gamma], modulo graph isomorphism, with \gamma a monomial in the half-edge \psi’s and vertex \kappa_j’s for j\geq1. At each vertex its decoration degree is at most 3g_v-3+n_v. The total degree is |E(\Gamma)|+\deg\gamma, and

There is no automorphism denominator in this definition of q_{g,n}. The per-vertex degree bound is the explicit convention in [PPZ, Section 0.3].

We use the usual decorated-graph intersection product. More explicitly, take common refinements with specified contractions to fixed representatives of the input graphs, requiring every refined edge to survive in at least one contraction. Divide by the source symmetries preserving those contractions. For each additional use of an overlapping edge insert its normal factor -\psi-\psi'; pull the leg and flag decorations to their components and split kappa classes additively. Specified contractions record the lifts to the gluing parameter spaces, so this convention gives the product of the unnormalized pushforward generators. The construction is the transverse and self-intersection rule for node cuts [GP03, Appendix A, Proposition 9 and Equation (11)].

Terms exceeding any vertex dimension are set to zero. For multiplication and restriction to a further graph splitting, an excessive decoration remains excessive: the combined dimensions on the pieces decrease, so at least one piece still exceeds its dimension. Forgetting pullback of an undecorated graph is obtained by adding the new leg successively at each vertex. These conventions lift the corresponding Chow operations and do not impose any unproved relation. The preceding dimension argument is not an assertion about forgetting, which raises a vertex dimension. Compatibility of forgetting with the formal detector is checked separately in Section 3.

2.2 The exact Pixton span

Define

For a commuting parity variable \zeta^2=1, put

Indices congruent to 2 modulo 3 are not used. For a series F write

Give every vertex of \Gamma its own variable \zeta_v, with \zeta_v^2=1. Define the operator

The empty product maps to 1, and the operator acts coefficientwise in T. The permutation-cycle expression is Faber’s formula, as recorded in [AC, Equations (1.12)–(1.13)].

For an edge whose two half-edges lie at v,w, with cotangent symbols x,y, set

This is a power series by

The same rule applies to a loop.

No converse to Proposition 2.2 is used. In particular, same-vertex multiplication of a relation does not enlarge the span, whereas an arbitrary new Chow relation is not assumed to belong to it.

2.3 The corrected cubic and its alternation

For the rest of the construction fix

Form a word \mathbf i in the colors 1,\ldots,g by concatenating all increasing triples of colors in lexicographic order. Its consecutive three-slot blocks will be denoted J. Each color occurs \binom{g-1}{2} times.

For a set of labels T of size at least two, let D_T be the one-edge class cutting off a genus-zero tail with precisely those labels. For distinct labels a,b, put

We also need a class supported on rational bridges, not tails.

Every operation in Definition 2.3 is finite. Each cubic factor has codimension two, so

The pair (g,n) is stable. The automorphism weights in \beta_J^{\mathrm{st}} specify coefficients of this particular class; they do not change the unnormalized generators of \Salg_{g,n}.

The proof of Theorem 1.1 now divides into two independent claims. Proposition 3.1 proves formal nonmembership. The remaining sections establish the geometric vanishing in Theorem 8.4.

3 A formal detector

We first separate the candidate from the relation space. Throughout this section the genus g is fixed as in Definition 2.3, but the number of labels is allowed to vary, including zero. No Chow relation on a positive-genus vertex will be imposed in the construction.

The maps are tests on formal strata. In particular, they are not asserted to factor through the Chow ring.

3.1 An isotropic cubic Frobenius algebra

The state space has the curve pairing familiar from Frobenius-algebra and CohFT constructions [Abr, PZ]. The exterior coefficient ring below allows an independent cubic on every triple of colors. We check associativity and the handle formula directly, since these particular identities govern the graph test.

Let

with each generator odd. Let W be a purely odd symplectic vector space of dimension 2g, with complementary Lagrangians W_+ and W_-. Choose bases a_1,\ldots,a_g and b_1,\ldots,b_g such that \langle a_i,b_j\rangle=\delta_{ij}. Extend the symbols \theta_{ijk} alternately to all triples of indices. There is an alternating, odd-coefficient three-form \Phi on W defined by

Consider the free supermodule

where 1,e are even. Its trace, denoted by \int, takes e to 1 and is zero on 1,W. Give V the even pairing which is hyperbolic on \Q1\oplus\Q e, the prescribed symplectic pairing on W, and zero between these two summands. Declare 1 to be the unit, e^2=eW=0, and determine the remaining products by

Equivalently, the product of two standard odd vectors is its symplectic pairing times e, plus the vector dual to the cubic functional in the third input. The latter vector has odd coefficients and belongs to R\otimes W_-.

Write \mu for multiplication and \Delta for its adjoint coproduct.

Set

For a nonempty finite label set L, write \Delta_L:V\to V^{\otimes L} for the coproduct adjoint to iterated multiplication; for a singleton it is the identity. Coproducts are coassociative and satisfy the projection formula. In particular, writing \Delta_{ij}(1) with units in all other slots, we have

The minus sign in (3.2) will match the normal bundle of a rational-tail boundary divisor.

3.2 Compression of rational tails

Call a graph a rational-tails graph if it is a tree with one genus-g vertex, called its root, and all other vertices rational. Every branch away from the root has nonempty labeled descendants. Such a graph has no automorphisms fixing the labels: the descendant label sets distinguish its branches, recursively.

We initially work without the degree cutoff, discard all graphs which are not rational-tails graphs, and allow genuine genus-zero Chow identities on rational parameter factors. This is only an auxiliary computation; we will prove that its output respects the cutoff in \Salg_{g,m}. We use the genus-zero Chow boundary presentation of [Keel, Section 4, Theorem 1].

For each label i, put

If the labels are ordered 1,\ldots,m, let L_x be the pullback of \psi_x from the space retaining just 1,\ldots,x. Thus

For j\geq1, define

The leading, graphless change of variables is \bar\psi_i=\psi_i and \bar\kappa_j=\kappa_j-\sum_i\psi_i^j, which is an invertible polynomial change. Hence the bar monomials form a complement to the boundary subspace before truncation.

Define t_m on that complement by

Here the tensor on the right has a unit in every unused slot. For a nontrivial rational-tails graph, integrate each non-root decoration on its genus-zero parameter space, retaining only its top-degree integral. Apply t_a to the decorated root with its a incident slots. Finally duplicate each root slot into the descendant labels of its branch by \Delta_L. Every branch attached to the root has at least two descendant labels, so a<m; this is a recursive definition. For m=0, the empty tensor is R, the unit maps to 1, and positive bar-kappa monomials map to zero.

3.3 Boundary operations and the test

We first show how the test treats multiplication by bar classes and rational-tail divisors. We then prove the forgetting identities that will reduce the number of labels in the relation argument. All tensor products, contractions, and permutations below use the super sign convention.

3.4 Forgetting markings

The forgetting identities are needed in both directions: pullback will remove a zero-index leg from the Pixton recipe, and pushforward will test a tensor slot against the two even basis vectors.

3.5 The zero-leg identity

We require one precise identity for the recipe in Definition 2.1. Package the partition insertions with formal variables s_\ell, one for each allowed positive part, using exponential coefficients for repeated parts, and put

The variables s_\ell are purely formal; all subsequent coefficient extractions involve finitely many of them. The logarithm below is expanded jointly in T and the s_\ell, so its constant term at T=s_\ell=0 is defined even though H(0,\zeta) depends on the partition variables. The cycle exponential formula for \widehat\kappa_\Gamma rewrites the vertex characteristic as

Here a monomial T^d\zeta^\epsilon in -\log H is replaced by \kappa_dT^d\zeta_v^\epsilon, including d=0. For the edges use U(x,x')=T\Delta_e, so the coefficient of total degree T^r is the degree-r recipe. Denote this packaged series, with its usual parity extractions, by \mathcal R(s;a).

3.6 Annihilating the relation span

3.7 The nonzero value on the candidate

We finish the proof of Proposition 3.1. Every graph in a bridge correction \beta_J^{\mathrm{st}} has nontrivial unmarked stabilization and is not a rational-tails graph. Its refinements, pullbacks, and products remain outside the support of the test. By (3.7) and (3.1), the remaining part of each factor of Y tests as

Pair slot j with the first-Lagrangian vector a_{i_j} prescribed by the color word in Definition 2.3. A block with increasing colors i<j<k gives \theta_{ijk}. The product of the unsymmetrized factors therefore evaluates, up to one fixed super sign, to

with the product ordered as the blocks.

Let G_{\mathbf i} be the stabilizer of the word. For \tau\in G_{\mathbf i}, transporting its super permutation across the pairing of odd input slots contributes \operatorname{sgn}(\tau); the input word itself is unchanged. This cancels the explicit sign of that summand in Y. No reordering of the exterior generators is required in this comparison: all summands are compared with the same ordered evaluation. Every color occurs \binom{g-1}{2} times, and hence

Together with Lemma 3.7, this proves the proposition.

4 Faithful tensor tests in relative Chow

The formal detector and the Chow-vanishing argument use different coefficient algebras. For the latter we must retain all Chow correspondences, including those not generated by the pairing. We construct such a tensor model in this section. Its coefficient algebra is generally large; the point is its faithfulness, not a presentation of its generators.

Let

be a smooth projective family of connected curves of genus b, with a section, over an irreducible smooth quasi-projective complex variety. Fix a rational divisor class e_C\in\CH^1(C) of relative degree one. The class e_C need not be the class of the section. Set

Here and below products of curve families are over S.

For an integer a>0, we may work modulo classes that vanish after restriction to S\setminus Z, for some closed subset Z\subset S of codimension at least a. This convention does not mean taking numerical equivalence, nor does it mean testing individual geometric fibers.

4.1 Correspondences and the curve projector

We use the rational correspondence category whose objects are smooth projective S-families and whose morphisms are

with all codimensions allowed. Composition is pullback to the triple product, intersection, and pushforward. Tensor product is relative product. Equivalently, this is the usual correspondence calculus with Tate twists forgotten. We take finite direct sums and split idempotents. The diagonal identifies every generating object with its dual; the evaluation for the curve is its intersection pairing followed by \pi_*.

The morphisms that vanish away from a closed base subset of codimension at least a form a tensor ideal. Indeed, a finite union of such subsets has the same codimension bound, and restriction to its complement commutes with every operation just described. We can therefore first quotient by this ideal and then split idempotents. Morphisms between the original objects are unchanged by the latter completion. Denote either the unquotiented category or this completed quotient by \mathcal C. Its tensor unit is the base S. The endomorphisms of that unit are \CH^*(S), or their base-support quotient; positive-degree base classes remain part of the correspondence calculus unless the chosen quotient kills them.

The projection formula gives

Thus 1_C,e'_C span a nondegenerate hyperbolic plane in the curve object. The two corresponding projectors sum to e'_{C,1}+e'_{C,2}, so their orthogonal complement has the self-transpose projector \eta. Write this complementary object as H_C.

The categorical trace of the curve identity is \pi_*c_1(T_{C/S})=2-2b, by diagonal self-intersection. Each of the two unit summands has trace one. Consequently

The remaining relation needed to model H_C by an odd symplectic space holds already in relative Chow.

4.2 Retaining arbitrary Chow correspondences

Let \mathcal R_b=\operatorname{Rep}^{\mathrm{odd}}(\Sp_{2b}). This is the usual semisimple rational representation category of the split symplectic group, with parity prescribed by its central involution. Its standard representation W_b is odd. The ordinary alternating symplectic form is therefore a symmetric pairing in the super sense; its closed loop has value -2b.

The first and second fundamental theorems for symplectic tensors present this category, after finite direct sums and idempotent completion, by pairing diagrams with that loop value and the sum-of-matchings relation on 2b+2 slots. In this formulation crossings on the standard odd space are minus the ordinary swap. See [RW, Theorems 2.2 and 2.13], or [LZ, Theorems 4.6, 4.8, and 5.9]. The diagram spaces and relations are rational, so the complex formulation descends by faithful scalar extension to \Q. Complete reducibility, absolute simplicity, and tensor generation over \Q follow from [Mil, Theorems 4.14, 22.2, and 22.42, and Paragraph 22.4]. In particular, the irreducibles occur as rational direct summands of tensor powers of the standard representation. Equations (4.4) and (4.5) thus give a symmetric tensor functor

For b=0, (4.5) says \eta=0, which gives the same conclusion with W_0=0.

It is essential not to assume that (4.8) is full. The target contains the small diagonal and many other correspondences. We record them by enlarging the coefficient algebra. The following reconstruction is a concrete split semisimple form of O’Sullivan’s tensor realization [OS, Lemma 2.3.1(i)]. It applies to a symmetric tensor functor with this semisimple source and an arbitrary rational tensor target; the endomorphism ring of the target unit need not be a field. We give its coefficient construction explicitly.

Apply the lemma to

In particular, a pure Chow class on C^k is a morphism 1\to H_C^{\otimes k}, so it is identified with an invariant in \mathfrak B\otimes W_b^{\otimes k}. This identification retains the class itself, including any component invisible to cohomology. The small diagonal and the structure map supply multiplication, unit, and integration. Their correspondence identities give a supercommutative Frobenius algebra on \mathfrak B\otimes V_b. The pairing is the one fixed by (4.3). In particular

is the tensor of the small diagonal after projection by \eta in all three slots. Its coefficients are odd, and it is alternating in the ordinary underlying vectors w_i. For a pointed curve over a field, this projection is the modified diagonal of Gross and Schoen [GS, Equation (0.1) and Sections 2–3]. The normalization here also incorporates the base corrections required for a family.

4.3 Colored operations and finite polarization

For an ordinary rational color space U, define

The product is pullback to disjoint curve slots followed by the coinvariant projection. With parity k\bmod2 in arity k, the direct sum of these spaces is supercommutative: interchanging blocks of k and \ell slots has sign (-1)^{k\ell}. The same definition applies to a collection of configuration spaces with equivariant forgetting maps, using their forgetting pullbacks for the product. We will use that version for weighted pointed curves.

A symmetric three-slot class gives an odd alternating entry f(i,j,k), multilinear in its colors. For the pure small diagonal, its scalar-color evaluation is \varphi(z(i),z(j),z(k)). We use a fixed ordering of tensor factors throughout; reordering several entries introduces exactly their Koszul signs.

5 A surface relation from determinant-fixed Quot schemes

The tensor formalism of Section 4 does not by itself bound products of the odd coefficients of a projected small diagonal. We obtain the additional relation we need from the surface square of the curve. The decisive features are a determinant constraint, which removes the surface obstruction directions, and a marked Hilbert-scheme calculation, which retains the Chow classes of all output positions.

5.1 The surface bound

Let \pi_C:C\to S be a smooth projective family of connected curves of genus b, with a section, over an irreducible smooth quasi-projective complex variety. Choose a rational divisor e_C of relative degree one, and use the corrected divisor e'_C, projector \eta, and odd alternating cubic form \varphi of Proposition 4.1. Set

Throughout this section, the support quotient is the quotient by Chow classes that vanish after deleting a closed subset of S of codimension at least b-1. Its coefficient superalgebra will be denoted by \mathfrak B=\mathfrak B_{S,b-1}.

Isotropic tests alone do not separate arbitrary Chow tensors. In Lemma 6.4, we first fix an arbitrary color map, find a subspace of its domain with isotropic image, and use the exterior-ideal argument to annihilate the entire sufficiently long product, throughout modulo base support of codimension at least b-1. Only after doing this for every color map do we apply the faithful test of Proposition 4.1.

We prove the localization relation underlying this theorem in the present section. Section 6 proves that its scalar coefficient is nonzero and thereby completes the proof.

Write

We write line bundles additively. On every fiber, K is ample, K^2=s_K, and \chi(\mathcal O_X)=\chi. The curve section gives a section o:S\to X. The two summands of the target bundle and their torus weights are

Here the use of w_1,w_2 for weights is confined to formulas concerning the target bundle; the tensor vectors retain their indices \alpha. Let \mathcal Q be the relative Quot scheme of rank-zero quotients

with the following determinant and numerical data:

All intersections in this display are fiber intersections. One may use the relative polarization K to specify the Hilbert polynomial. Projective representability and the Grassmannian construction are supplied by Grothendieck’s Quot theorem [Nit, Theorem 5.1 and Section 5]. The determinant condition is a closed fiber of the separated relative Picard scheme [Kle, Theorem 4.8(1)], so \mathcal Q is projective over S.

5.2 The derived obstruction theories

The determinant constraint changes both the obstruction space and the virtual dimension of \mathcal Q. We first construct its tangent complex as the fiber of the determinant differential. The surjectivity of that differential on obstruction spaces will give a two-term theory. The same construction on the torus-fixed factors will then express their virtual classes as virtual pullbacks from one rank-one Quot scheme.

5.2.1 Arrows and the Quot tangent

The perfect-complex moduli theorem of [TV, Corollaries 3.15, 3.17, and 3.19, Definition 3.18, Lemma 3.27, and Corollary 3.29] applies over every affine open of S. Indeed, the dg-category of perfect complexes on a smooth proper scheme over an affine base is proper by perfection of relative derived Hom and smooth by perfection of its diagonal. These smooth proper categories are saturated in the sense of [TV, Lemma 3.27 and Corollary 3.29]. Their moduli of perfect objects and perfect arrows are locally geometric and locally of finite presentation. These constructions glue over S. The tangent of the object moduli at a perfect complex F is \mathbf R\Hom(F,F)[1]. All Hom complexes in what follows are derived Hom on the surface, followed by derived pushforward to the parameter scheme. Perfect direct images and derived base change are used in the form of [Stacks, Tag 0B91], and relative duality in the form of [Stacks, Tags 0BRT, 0A9Q, and 0B6I].

5.2.2 Fixing the relative determinant

Let \mathcal P_o^{\mathrm{der}} be the derived line stack rigidified along o, namely the fiber of evaluation along o over the trivial line. For any line M on a base-changed surface, its normalization is

It has its canonical trivialization along o. Because (\pi_X)_*\mathcal O_X=\mathcal O_S universally, all line automorphisms come from the base, and the normalization removes them. The classical truncation of \mathcal P_o^{\mathrm{der}} is therefore the relative Picard scheme, parametrizing lines modulo base twists. Its tangent is

where constant functions split evaluation \mathbf R\pi_{X*}\mathcal O_X\to\mathcal O_S. This complex has geometric-fiber cohomology H^1(\mathcal O_X) in degree zero and H^2(\mathcal O_X) in degree one.

The determinant of perfect complexes is a functorial derived morphism [STV, Section 3.1]. Its differential is trace [STV, Proposition 3.2]. After normalization it maps the open arrow enhancement to \mathcal P_o^{\mathrm{der}}. Take its derived fiber at the normalized line L_1+L_2-D. Its truncation is exactly \mathcal Q: it fixes the Picard class of \det I', not a trivialization of the unnormalized determinant on a parameter scheme. The relative tangent restricted to \mathcal Q is

Here the trace identification is an identity of relative complexes, not an inference from equality at closed points. Locally represent a perfect family by a bounded complex P^\bullet of finite free modules. Its determinant is \bigotimes_j(\det P^j)^{(-1)^j}. Differentiation on infinitesimal automorphisms gives the supertrace \sum_j(-1)^j\mathrm{tr}_{P^j}. The same calculation with square-zero complex coefficients gives the chain map \mathbf R\Hom(P^\bullet,P^\bullet)\to\mathcal O; it respects homotopies and change of perfect resolution. Using the identification of object tangents with shifted loop tangents gives its shift by [1] as the determinant differential. This construction glues and commutes with base change, and therefore gives the relative trace in (5.7), including over nonreduced parameter schemes.

Applying Lemma 5.3 with r=2 and D=3K shows that the only possible degree-two cohomology of (5.7) vanishes. The complex is perfect of Tor-amplitude [0,1]. The canonical map from the cotangent of a derived enhancement to that of its classical truncation has cone in degrees at most -2 [STV, Proposition 1.2 and Corollary 1.4]. Consequently the dual of (5.7) is a relative perfect obstruction theory in the sense of [BF, Definitions 4.4 and 5.1]. Adding the tangent of the smooth base by transitivity gives the absolute theory. All constructions are torus equivariant.

5.2.3 Virtual rank and compatible pullbacks

Riemann–Roch gives the relative virtual rank

Indeed, if q_2=\int\ch_2Q, then \chi(V,Q)=2q_2-(K+L_1+L_2)D and \chi(Q,Q)=-D^2, while the rank of (5.6) is 1-\chi. Thus fixing determinant contributes the extra \chi-1 in (5.8). The inequality v>2t_0 will force the marked localization relation.

To compare the fixed virtual class with simpler classes, we need morphisms of obstruction theories, rather than only tangent classes in K-theory. The derived enhancements supply these morphisms as follows. For a morphism of these derived enhancements, transitivity maps to the transitivity triangle on truncations. If the two absolute comparison cones are C_Y,C_Z, the relative comparison cone fits into

The absolute cones lie in degrees at most -2; derived pullback preserves this upper bound and [1] lowers it. Therefore C_{Y/Z} also lies in degrees at most -2. Whenever the relative derived cotangent has perfect amplitude [-1,0], this diagram supplies the compatible triple of obstruction theories. Virtual-pullback functoriality then applies [Man, Definition 4.5, Theorem 4.8, and Corollary 4.9]; see also [STV, Proposition 1.5].

5.3 Fixed loci and their virtual classes

The equivariant embedding and resolution hypotheses of virtual localization [GP, Section 1] hold here. The Grassmannian construction of relative Quot is equivariant and embeds the determinant fiber in a smooth scheme over the smooth base S. One can construct a global equivariant two-term presentation of its perfect obstruction theory using a linearized ample line. For completeness, over an affine base sufficiently positive twists generate its degree-zero cohomology and kill the higher cohomology of its degree-minus-one cohomology. Lifting generators gives a bundle mapping to the complex and surjective on degree-zero cohomology; its derived fiber is a bundle by the Tor-amplitude criterion. Further Serre vanishing lifts the map to the two-term embedding cotangent presentation. Torus linear reductivity makes the choices equivariant. For a nonaffine S, the same argument may be made after a Jouanolou affine-bundle pullback [ADO, Proposition 4.1.4(1)]. The resulting identity descends by equivariant Chow homotopy invariance and smooth pullback of the base-changed obstruction theories. Subsequent dimension arguments are on the original base S.

A torus-fixed subsheaf of L_1\oplus L_2 splits into its two weight subsheaves, also in families. Kernels and quotients in the two weights are flat, since they are direct summands of flat families. Thus a fixed point is a pair of rank-one arrows, subject to the total determinant constraint. Their direct-sum morphism of derived enhancements identifies its tangent triangle with the weight-zero part of (5.7), compatibly with the cotangent comparison. Normalizing a determinant cancels all character lines pulled from the base, so the based condition does not acquire a moving term. The moving virtual normal complex is

with weight w_j-w_i.

We use the divisor-and-residual-point factorization established in the relative setting by [Fog, Theorem 1.4]. We spell out its compatibility with the determinant and nonreduced bases. Each rank-one kernel has a unique factorization

Here \mathcal C_i is a relative effective Cartier divisor and Z_i is a finite-flat point family. This holds over nonreduced parameter schemes. To see this, let T be such a parameter scheme and untwist the target line. The T-flat kernel ideal I has a length-one locally free resolution, by the fiberwise projective-dimension bound. Its determinant is a line bundle on X_T, commuting with arbitrary base change. The maximal-minor map I\to\det I restricts on every surface fiber to the inclusion of a rank-one torsion-free sheaf into its double dual. It is therefore fiberwise injective with zero-dimensional cokernel. Since both its source and target are T-flat, the local flatness criterion makes the map injective and its cokernel \mathcal T T-flat. Properness and the fiber support make \mathcal T finite over T.

The inclusion into the original target line extends uniquely across that finite support: on a smooth relative surface a finite-flat module has grade at least two, as follows by lifting a fiber regular sequence with flat successive quotients. Thus its Hom and first Ext into the structure sheaf vanish. Applying Hom to 0\to I\to\det I\to\mathcal T\to0 gives the extension. The extension is fiberwise an injection of lines with Cartier quotient; the local flatness criterion makes its divisor relative Cartier. It gives the required point ideal and divisor, uniquely and compatibly with base change.

Let the point lengths be r,s. The Chern character of the two quotients gives

The fixed scheme has only finitely many numerical choices. Its summand Hilbert polynomials are locally constant. In addition, the determinant line of each perfect kernel is a base-change-compatible line family, so its fiber Hilbert polynomial is locally constant. The residual point length is the difference between the Euler characteristics of this determinant and its kernel, and is therefore locally constant as well. Finite type now gives finiteness of these choices. All divisor and Picard moduli used in a choice may be restricted to the required finite-type quasi-projective pieces. For instance, lines of a fixed Hilbert polynomial form an open in the moduli of stable rank-one sheaves; rigidification at o represents their Picard classes.

Choose a high index with KD_i>K^2, which exists since KD_1+KD_2=3K^2, and call the other index low. Write P=D_{\mathrm{low}}. For this numerical choice, denote the fixed locus by \mathcal F and the unrestricted low rank-one Quot scheme by \mathcal Q_{\mathrm{low}}. Forgetting the high arrow gives a map

Its relative derived tangent is

Lemma 5.3, now with rank one, gives amplitude [0,1]. The transitivity comparison above is therefore a compatible triple. Consequently

Here f^! is a bivariant virtual pullback. We will use its action on cycle representatives of the low virtual class, including cycles supported on special divisor loci.

Obstruction Euler factors of point Hilbert schemes also enter the surface Quot calculations of Oprea and Pandharipande [OP, Lemma 34 and Section 5.2]. We establish the relative comparison with varying determinant and arbitrary irregularity needed here.

Let

be the smooth projection. Combining point excess with (5.12) gives the fixed-cycle formula

This formula separates the two remaining tasks. We first determine where the low divisor virtual cycle can be supported. Over the surviving support, the high map f becomes smooth, and its virtual pullback will supply the second point Euler factor.

5.4 Support of the low divisor class

Put y=KP/s_K. The virtual relative dimension of the divisor scheme is

If 0<y<1, this is at most -(b-1). To check the integer bound, put q=b-1. The positive intersection p=KP is a multiple of 2q: each of the two canonical divisors on the product has this fiber degree. Thus

by Hodge index and convexity in p. The left side is integral, proving the claim. The absolute divisor virtual cycle consequently has a representative of dimension at most \dim S-(b-1). The image of its support in S has codimension at least b-1. In (5.14), smooth pullback by \rho and cap product with e(F_{\mathrm{low}}) preserve this base support. So does f^!: it is bivariant and commutes with restriction and proper pushforward [Man, Theorems 4.1 and 4.4]. Although these pullbacks can increase the dimension of a cycle, they do not enlarge the locus over which it lies in S. Every marked fixed contribution above this low cycle therefore vanishes in our support quotient.

On the Picard open where H^2(P)=0, consider instead the morphism from the divisor enhancement to the derived based Picard scheme. The divisor exact sequence shows that its obstruction map surjects onto H^2(\mathcal O_X). Hence its relative tangent has amplitude [0,1], and the transitivity construction again gives a compatible virtual-pullback triple. The absolute virtual dimension of that Picard theory is

Its virtual class is zero, so the restricted divisor virtual class is zero as well. For y>1, the condition H^2(P)=0 is automatic: otherwise duality makes K-P effective, contrary to ampleness. For y=1, a nonzero section of K-P has zero intersection with K, and therefore has empty divisor. Thus the exceptional line is exactly K. Chow localization expresses the divisor virtual class as a pushforward from the canonical linear system |K|. For y=0, an effective divisor is empty; its tangent complex is zero and its class is the base fundamental class.

We have reduced the low divisor support to 0 or K, and hence the high line to 3K or 2K. On a neighborhood of either support the high direction is smooth: Kodaira vanishing gives its projective bundle of sections, together with its smooth point Hilbert scheme. The relative virtual tangent class there is

Indeed, the unrestricted rank-one arrow class subtracts \mathbf R\pi_*\mathcal O_X, while fixing the based determinant subtracts \mathbf R\pi_*\mathcal I_o[1]; together these subtract exactly one. The remaining point-ideal calculation is that in Lemma 5.4. The projective Euler sequence identifies actual tangent minus (5.16) with [F_{\mathrm{high}}].

For a smooth morphism with a relative perfect obstruction theory, the obstruction sheaf is locally free and its virtual pullback is smooth pullback capped with its top Chern class. This follows by locally splitting the cotangent comparison onto the locally free relative cotangent; equivalently, the intrinsic cone is [0/T_{\mathrm{rel}}] and zero-section excess gives the Euler factor. The argument works on arbitrary base cycles, not just the fundamental cycle; compare [BF, Proposition 7.3] and [Man, Theorem 4.4]. Here the obstruction bundle has K-class [F_{\mathrm{high}}], so its Euler class is e(F_{\mathrm{high}}). Let \mathcal D\subset\operatorname{Div}_{P}(X/S) be the empty-divisor locus S or the canonical linear system |K|. Base-change f to \mathcal D\times_S X^{[r_{\mathrm{low}}]}_{/S}, and use the same letters f,\rho for the resulting maps. For every cycle \alpha on this product, the high virtual pullback is therefore

In particular, apply it to \alpha=e(F_{\mathrm{low}})\cap\rho^*\beta, where \beta is a cycle on \mathcal D supplied by Chow localization for the low divisor virtual class. Bivariant proper-pushforward compatibility then returns the result to \mathcal F. All operational insertions restrict to the actual linear systems on those cycles; no formula for the low cycle supported on |K| is needed. These linear systems are smooth over S: for K, constant cohomology dimensions follow from the product of curves and base change, and for 2K,3K from Kodaira vanishing. Their universal divisor classes are D_i+x_i, where x_i is the hyperplane class.

5.5 Marked localization and the short-length exclusion

Adjoin m=3t_0 independent surface positions to \mathcal Q. At each position insert \ch_2^{\mathbb G_m}(Q), cap with the smoothly pulled virtual class, and push to X^m_{/S}. This push is proper. The torus acts trivially on the output. Before localization the total codimension is 2m-v. An ordinary output component of codimension 4t_0 would therefore have equivariant exponent

It is absent. Thus the ordinary codimension-4t_0 part of the localized fixed sum is zero after setting the equivariant parameter equal to one.

On a fixed branch the insertion is a sum, for i=1,2, of

Here [Z_i] denotes the incidence cycle on the corresponding surface position. Test every surface slot by the even tensor

This is a tensor test in the coefficient algebra, not a numerical intersection test alone. Since a unit curve class pairs to zero with a pure W_b input, a line-only insertion that survives this test uses a positive divisor degree on both curve axes. Hence each such position supplies at least two ordinary codimensions, from powers of the two curve canonical classes.

Suppose k of the positions use line-only insertions. Extract these 2k divisor codimensions by the projection formula. The remaining m-k incidence positions lie in the union of the two point subschemes, of total length r+s. Their pushed support therefore has at most r+s distinct surface positions and has codimension at least

in those factors. Operational incidence classes vanish away from their incidence supports, so this support assertion applies to successive caps with arbitrary low cycle multipliers. If r+s<t_0, the displayed codimension is strictly larger than the available 4t_0-2k, and the tested contribution is zero.

By (5.11), the four remaining ordered divisor pairs have lengths

Only (D_1,D_2)=(0,D) survives. On this branch the low divisor is empty and its virtual class is the base class. The calculation is on the product of the two point Hilbert schemes and |D|, with both point Euler factors and the ordinary fundamental cycle. We have

5.6 A relative marked Hilbert-scheme recursion

The surviving fixed contribution must be expanded as a Chow class on X^m_{/S}, with every output retained. Relative EGL recursion for surface pushforwards to a base appears in [Kong, Section 2.1, Lemma 2.1 and Proposition 2.2]; the same work also treats one and two marked curve outputs [Kong, Section 3.3, Lemma 3.4]. The construction below retains an arbitrary finite set of surface outputs. This is the form that lets the codimension bound identify the Chow tensor left by localization.

Apply Lemma 5.5 to the surviving fixed branch. For the surface product of curves, all the allowed output decorations expand into powers of the two curve canonical classes. Consider a connected equality block containing k outputs. For k=1, pure-slot testing requires at least two decorating codimensions, one on each axis. For k=2, a bare diagonal on either curve axis pairs two isotropic inputs and is zero. Decorations are therefore needed on both axes, in addition to the codimension-two surface diagonal. For k\geq3, the equality block already costs 2(k-1) codimensions. These lower bounds are summarized in Table 2. Every surviving block costs at least 4/3 codimensions per output, with equality only for an undecorated three-output block.

The total selected codimension is 4t_0 on 3t_0 outputs. It follows that only undecorated triple blocks occur. No line-only insertion survives. There is no remaining positive horizontal codimension: every closed surface component must have top relative degree, and every base bundle contributes only its scalar characteristic term. Integration over |D| therefore extracts exactly x^{t_0}. In particular, the scalar multiplying the triple blocks can be computed over a point with the same numerical characteristic data, using the fixed polynomial procedure of Lemma 5.5.

5.7 The scalar interface

To specify this scalar without an independence assumption on diagonal classes, use the polynomial procedure just fixed. For a formal surface input z, write a connected component carrying k marked inputs and decoration \alpha as the moment

Give unmarked moments their actual top intersection numbers. Give marked moments value zero except for M_3(1)=1. Thus this evaluation retains precisely bare triple blocks. In the generating series of Section 6, the variable q records total point length, p is the exponential variable for marked positions, and x is the high hyperplane class. Let a_{N,t_0} be the coefficient defined in (6.4), with the moments evaluated in this way and N as in (5.20). Equivalently, (3t_0)!\,a_{N,t_0} is the sum of the coefficients of all bare triple partitions in the labeled marked pushforward. The polynomial procedure is fixed before imposing relations among the diagonal classes of a particular surface. Section 6 evaluates this same representative: its moment-additivity argument is what permits numerical surface calculations to determine the selected scalar. Thus the passage to those calculations does not replace the marked Chow identity by numerical universality.

6 The triple coefficient and arbitrary color products

We first compute the scalar left by Proposition 5.6, completing the surface bound. We then convert that bound on isotropic tests into vanishing for sufficiently long products with arbitrary color inputs. For the coefficient calculation, put

The letter q in the generating series below is a formal variable. It is unrelated to the realization maps q_{g,n}.

The calculation belongs to the study of tautological Hilbert-scheme integrals developed by Lehn and Ellingsrud–Göttsche–Lehn [Leh, Section 4.3], [EGL]. For the symmetric-power and residue methods used below, compare [MOP2, Equation (9), Theorem 2, and Lemma 3]. The coupling of two Hilbert factors, all marked moments, and the Euler-shift derivative will be retained explicitly.

We first identify a generating series for this scalar. We then compute its first nonzero hyperplane coefficient and prove that the resulting one-variable coefficient cannot vanish.

6.1 The moving factor

For a vector bundle E, write e_a(E) for its Euler class after adding a to each Chern root. If a has invertible constant term, this convention extends multiplicatively to virtual bundles and perfect complexes. Inverses are expanded in positive Chow degree. Let

with universal finite subschemes Z_1,Z_2 on M_{r,s}\times X. Derived Hom expressions in this subsection include pushforward along the surface X.

On the surviving fixed component, the two kernels are

where x is the hyperplane class of |D|. Apart from the factor (1-x)^{\chi(B)}, the inverse moving Euler class is

Here and below only scalar terms of bundles pulled from the original base can contribute, by Proposition 5.6. In particular the omitted inverse Euler factor of R\pi_*(-A_0), at weight one, has scalar term one.

To check (6.1), the forward moving arrow has virtual tangent

at weights 1 and 1-x, respectively. Expand each point ideal as \mathcal O-\mathcal O_Z. The constant term of the second Hom gives (1-x)^{\chi(B)}. Surface duality identifies

in the convention relevant to the Euler class, and likewise for -A_0. This gives the first numerator and the first two denominator factors of (6.1). The reverse moving arrow is

at weight -1+x. Its point term gives the final denominator. Duality changes its cross term to the remaining numerator term. The cross Hom has virtual rank zero, so changing both its dual and the sign of its weight introduces no rank sign.

The fixed obstruction Euler classes on this component are

The sign of the second shift is worth recording: the line in the obstruction bundle is K-D-x before taking the dual, and hence has twist +x afterwards.

6.2 Marked moments and shifted additivity

Temporarily replace K and K-D in these two fixed Euler classes by independent divisor variables L,H. Also separate the Euler shift x_e from the moving shift x_m. For a surface class z, let

and define \mu_s(z) similarly. Consider

The exponential records labeled output insertions with their usual factorial denominators.

We use the fixed marked recursion of Proposition 5.6 to regard each finite coefficient of (6.2) as a polynomial in decorated moments

Here \alpha is a monomial in L,H,A_0,D,K,c_2(T_X). More precisely, carry out the nested-Hilbert recursion while retaining the marked output slots. Each connected component of the final equality graph is integrated over one copy of X; if it contains k marked slots and decoration \alpha, replace that integration by M_k(\alpha). This fixes a polynomial representative before imposing numerical relations between moments. The recursion and its projective-bundle signs are those of [EGL, Sections 1 and 3]. The version with arbitrary retained outputs used here is proved in Lemma 5.5. This is the interface that keeps the calculation in relative Chow throughout.

Let \mathcal E denote the following evaluation:

Unmarked integrations in this formula are their top-degree numerical integrals; other degrees give zero. The scalar normalization agreed with Proposition 5.6 is

Indeed, \mathcal E retains precisely the partitions into bare triples. Passing from the exponential series to labeled outputs multiplies by the nonzero common factor (3t)!. Every such partition has the same tested value, up to the common Koszul sign for grouping the two curve axes. Those factors have already been divided out in Proposition 5.6.

The next observation allows us to compute (6.4) using ordinary numerical surface calculations, without assuming that all formal moments are independent.

6.3 Restriction to two curves

We first claim that

Represent very ample L,H by smooth transverse curves. At x_e=0, the two Euler classes in (6.2) restrict the integral to L^{(r)}\times H^{(s)}, with sign (-1)^{r+s}. These are the regular zero loci of the tautological sections: the zero schemes are the Hilbert schemes of points on the two smooth curves, of the expected codimensions r,s. A point-class perturbation of the constant z=u restricts to zero on both curves. The Vandermonde test in the preceding proof therefore annihilates the bare marked coefficient. The coefficient of the single moment M_3(1) is a universal scalar, independent of the numerical line classes, so this proves (6.7) identically.

To determine c_H, set both shifts to zero and take

for a divisor Z on the surface. Write x',y' for the numerical point divisors on L^{(r)},H^{(s)}. After the Euler restrictions, the one-body factors of (6.1), including their signs, are

Here c_a(E)=\sum_j a^j c_j(E). For verification, on the ordered cover of a symmetric power, a tautological line bundle M^{[r]} has a filtration with roots

Two line classes pulled from the same curve position multiply to zero. Applying this to the low ratio in (6.1) reduces its factor at position i to

which gives the first term in (6.8).

Put I=LH. At each of the I transverse intersections, the two universal finite subschemes meet by independently imposing their point-containment divisors. Their derived Hom has the class of the structure sheaf of the product of those divisors, twisted by the determinant normal of the first universal subscheme. Numerically its class is

Its Euler class at weight one is therefore

The two cross Hom terms give the combined factor

This calculation also follows by resolving the two transverse containment divisors and taking their tensor-product resolution; the twists by surface lines have numerical degree zero at the fixed intersection point. The incidence exponential becomes

We recall the symmetric-power integral needed to sum these terms. For a curve of genus h, a line of degree d_1, and its tautological bundle on the r-th symmetric power,

Here \vartheta is the pulled-back Jacobian polarization. For completeness, the universal divisor sequence and GRR give

Indeed the universal divisor class has terms r[\mathrm{pt}]+x'+\ell, where the mixed class satisfies \int_{\mathrm{curve}}\ell^2/2=-\vartheta. Converting this Chern character to the total Chern class gives the first formula in (6.11). For the second, pull to the ordered cover and expand the polarization in a symplectic basis. Intersecting the point divisors first leaves j factors; choosing j distinct symplectic pairs contributes one after the factorials, giving \binom hj.

Let

By (6.11), low-curve integration of the first factor in (6.8), times any series in x', is coefficient extraction against

High-curve integration has the analogous expression after replacing y' by -y', which absorbs the sign (-1)^s. Accordingly (6.9) becomes

and the nonconstant part of (6.10) becomes \exp(p\tau(ZL\,x'-ZH\,y')).

The elementary Lagrange identity used for each variable is

One proof is to express coefficient extraction as the formal residue at zero: summing the geometric series gives F(x)(1+x)/(x(1+x)-Q), whose small root is w; its residue there is the right side. The identity holds with other formal variables in the coefficients, so it can be applied successively to x' and y'. Using Q=q\exp(pu), both small roots equal w. Equations (6.12)–(6.13) therefore give

This equation is the genuine numerical evaluation with z=u+\tau Z. It determines the universal coefficients used below.

6.4 The first hyperplane coefficient

Let w_0(1+w_0)=q, with w_0(0)=0, and let \mathcal D=q\partial_q. By Lemma 6.2, the numerical evaluation of \log\Theta on a genuine moment vector equals that of its one-moment part: every component with at least two moment factors vanishes there. Its term of p-degree three is a universal scalar combination of the moments M_3(\alpha). The term involving M_3(H) is c_H(q)M_3(H)p^3/3!. Since

we can identify c_H by varying the very ample H, with the other divisor classes fixed. Only degree-one decorations can contribute to u^2\tau, and H is the only varying such decoration. For example, one may take arbitrarily large integral multiples of a fixed ample line for H, with Z fixed ample. The first two terms of (6.14) contain no \tau. The coefficient of p^3u^2\tau(ZH) in its last term is -\mathcal D^2w_0/2. Comparing factorials yields

Together with (6.6) and (6.7), this proves

There is no contribution from differentiating x_m, because c(q,0,x_m) is identically zero.

The unmarked calculation is equally explicit. Set p=0 in (6.14). Every finite coefficient is polynomial in the intersection numbers of L,H,A_0,D,K, so the formula continues from very ample L,H to L=K,\ H=K-D. This can also be seen by varying sufficiently large multiples of K in both variables and then using polynomial identity. The adjunction expressions for h_1,h_2 are polynomial expressions in this continuation, not assertions that a negative line has a smooth effective representative. Substitution gives

6.5 A residue with a definite sign

We now prove Proposition 6.1. By (6.5), the p^{3t}-coefficient is c(q,x,x)^t/(6^t t!) times the unmarked series. The series c(q,x,x) has zero constant term in x. Thus the coefficient of x^t uses its linear term in every factor, and uses only the constant terms in x of the unmarked series and of (1-x)^{\chi(B)}. Equations (6.4), (6.15), and (6.16) give

Put v=1+2w_0, so that q=(v^2-1)/4. The coefficient residue in (6.17) becomes

For the present parameters,

Hence the rational differential in (6.18) has finite poles only at zero and one. Its rational coefficient has degree 2-\chi\leq-14 at infinity, so its residue at infinity is zero. The residue at zero is

This is strictly positive: the second factor has positive coefficients in every nonnegative degree, the first has nonnegative coefficients and constant term one, and 27\chi-5\geq0. The residue at one is its negative and therefore is nonzero. Both scalar prefactors in (6.17)–(6.18) are nonzero. This proves a_{N,t_0}\neq0.

Finally, Proposition 5.6 gives a_{N,t_0}T_\gamma^{t_0}=0 for the isotropic cubic contraction T_\gamma appearing in Theorem 5.1. The scalar is a nonzero rational number and is invertible in the coefficient algebra. Thus T_\gamma^{t_0}=0, which completes the proof of that theorem.

6.6 From isotropic tests to arbitrary color products

The surface theorem now gives nilpotence for every isotropic tensor test. To use it in the product defining the counterexample, we need a vanishing statement for arbitrary color inputs. We return to a smooth genus-b curve family, choosing a rational divisor of relative degree one and its pure projector as in Proposition 4.1. We use the operation algebra \mathcal A_k(E) of Section 4 for an ordinary rational color space E. A pure cubic entry is the class of the projected small diagonal in \mathcal A_3(E) with a specified triple of color inputs; products use disjoint curve slots.

We will fix an arbitrary color map E\to W_b, apply the surface theorem to a subspace with isotropic image, and use the following exterior-algebra lemma to annihilate every sufficiently long product. The final application of faithful polarization will then give a Chow identity with the same base-support bound.

The smooth-family argument now gives vanishing of long products of pure cubic entries for arbitrary color inputs, modulo the stated base-support bound. The remaining task is to compare the compact candidate with these pure entries and control the errors along the boundary.

7 A cubic class on weighted compactifications

Lemma 6.4 gives vanishing of long products of projected small diagonals over smooth curve families, modulo the stated base support. To apply it on the entire moduli space, we construct a compact cubic operation whose restriction to a boundary chart differs from these pure diagonals by controlled terms. Weighted stable curves and reduction morphisms are due to Hassett [Has]. The smooth-core configurations below are related to the blowup compactifications of Fulton–MacPherson [FM] and their weighted relative versions [Rou]. For rational tails, Tavakol’s proposed comparison with powers of the universal curve [Tav] was established by Petersen [PetW, Proposition 2.11]. Here we also need control under cuts of nodal heavy cores. The correction at a rational bridge is essential: without it, a codimension-two exceptional class would remain on a three-light-point configuration space.

We distinguish three cubic objects. The formal class F_J^{\mathrm{st}} belongs to the strata algebra and enters the definition of Y. For the empty heavy set, the corrected weighted cubic F^0_{g,\varnothing} pulls back under weight reduction to q(F_J^{\mathrm{st}}), with the corresponding label-forgetting pull. We also use its flagged versions F^0_{h,B}, called raw only to distinguish them from the auxiliary pure lifts G_{h,B}. Each G_{h,B} is obtained from F^0_{h,B} by controlled corrections: it agrees with the projected small diagonal over the smooth-core open, not globally on the compactification.

7.1 Weighted configurations and cut charts

Let (h,B) be stable heavy data: B is a finite ordered set and 2h-2+|B|>0. Write

with X_{h,B}(0)=\Mbar_{h,B}. Here and below any positive rational weights of total at most one give the same moduli problem for the light labels. Lights can coincide with one another, but not with a heavy marking. A rational component with one node and only light markings cannot be stable; a rational component with two nodes and positive light mass is stable. These observations also show the independence from the individual light weights.

The stacks X_{h,B}(k) are smooth and proper, and reduction of weights and forgetting labels give the usual stabilization morphisms [Has, Theorem 2.1, Proposition 3.8, Theorems 4.1 and 4.3]. The one-light space is the universal curve over \Mbar_{h,B}: when its coarse position is a node or a heavy marking, a unique three-special-point rational component is inserted. Thus sections and the relative dualizing line define natural one-light classes.

Fix a stable graph \Gamma of the heavy type. At its vertex v let h_v be the genus and let B_v comprise the original heavy markings and the incident half-edges, all ordered. Put

The cut chart with k lights is the disjoint union

It has a proper gluing map to the weighted space of the original heavy type. Assignments are part of the indexing, so a class supported on the assignment of specified labels to a vertex means zero on the other components of (7.1).

Refining a graph gives further proper cut charts. These charts cover the preimage of the corresponding heavy boundary. Indeed, a node of the light-forgotten curve may have a chain above it before stabilization. Choose a node on that chain and cut it, replacing its branches by heavy flags. The resulting pieces remain weighted stable; gluing recovers the original curve. Several prescribed nodes can be cut independently. The map on heavy bases is finite, as for the ordinary gluing of stable curves.

Cutting and light forgetting commute. The heavy flag replacing a node branch retains exactly the stability contribution of that branch. Thus stabilizing on the separate pieces and then gluing gives the same curve as forgetting and stabilizing after gluing. This compatibility will permit operations on unconsumed light labels to pass through any later cut.

We use rational Chow classes operationally on these smooth stacks. For an ordinary color space U, the arity-indexed operation algebra is

The product pulls operations to disjoint sets of labels by forgetting and then multiplies them. Successive stabilization proves associativity; exchanging the two sets of labels gives the sign (-1)^{k\ell}. In particular, cubic and singleton operations are odd and pair operations are even. All expressions below are first identities of labeled classes and then identities in (7.2).

7.2 The corrected cubic

For three light labels define

The coincidence class is the product of the divisors [1=2] and [1=3]. The class \beta_{h,B} is defined by taking each stable one-edge graph of the heavy type, subdividing its edge by a genus-zero vertex carrying exactly the three lights, and summing the resulting gluing classes divided by their graph automorphism orders. The divisor \delta_{p,123} has a tail containing exactly the heavy mark p and the three lights; \psi_p is the heavy cotangent class on the weighted space. Each term in (7.3) is symmetric in the three light labels.

7.3 The local three-position calculation

Over \mathcal S_\Gamma let C_v denote the smooth core curve at vertex v. For a fixed assignment of three labels, there is a coarse position map

All products in this subsection are over the indicated heavy base.

Denote the total first exceptional divisor at a triple flag locus by E_{p,123}. Denote pair exceptional divisors, preferably pulled back from the corresponding two-light spaces, by E_{p,ij}. The blowup formula for Chow groups [Ful, Proposition 6.7(e)] shows that a codimension-two class on (7.4) has, in addition to its main pullback, terms of the following forms:

Here A_{p,ij} is a divisor class on the remaining curve over the base, D_p is a base divisor class, and a_p is a scalar on each connected component of the base. For clarity, if E_{p,ij}^{\mathrm{str}} is the strict pair exceptional divisor and E_{p,ij}^{\mathrm{pull}} is pulled back from two labels, then

Multiplying the difference by a divisor on the remaining curve restricts that divisor, on E_{p,123}, to the flag section. It is therefore a pulled base divisor. Replacing the strict version by the pulled version changes only the triple-exceptional-times-base-divisor term, and introduces no new scalar multiple of E_{p,123}^2. Formula (7.5) includes assignments to different vertices, in which some of the centers are simply absent.

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 is zero otherwise. Away from flags this is immediate. The coincidence, bridge, and heavy-tail corrections not seen there project to loci where all three coarse positions are flags, of codimension three; they cannot alter a main codimension-two class.

For the same pulled raw class, the divisor coefficients A_{p,ij} in (7.5) are supported on flag sections of the remaining curve, up to base divisor terms. To see this, remove all flags from that remaining curve. Neither the raw class nor its main diagonal pullback meets the pair exceptional locus over this open. Intersecting with the exceptional divisor and pushing to its center recovers its coefficient, with the usual negative self-intersection sign for a codimension-two blowup center. The coefficient is therefore zero on this open. Chow localization on the curve over the base gives the assertion.

7.4 Controlled pure lifts

For a smooth heavy core \pi:C\to S, choose a degree-one rational class

The second case occurs only for h\geq2. As in the curve tensor construction, put

Write \Delta_C^{\mathrm{pure}} for the small diagonal projected by \eta_C in each of its three slots. Expanding the three projectors shows that

No numerical equivalence is used in this expansion: it is the decomposition of the diagonal correspondence into the pure projector and the two hyperbolic projectors.

8 Support induction and Chow vanishing

We now combine the arbitrary-color bound of Lemma 6.4 with the compact lifts of Proposition 7.4. We will make successive parts of the complete product of three-color entries vanish over dense opens of its base support, until no support remains. Throughout this section,

The basis of U_0 is the set of colors. An entry is one factor with three specified color inputs. Initially there is one entry for each increasing triple of distinct basis colors. Products and their factorizations are taken in the sign-coinvariant operation algebra of the weighted spaces from the preceding section.

The graph charts, their compact heavy bases, and their smooth-core opens will be denoted by

respectively. Every graph in this section has unmarked type (g,0); its vertex flags are heavy marks. An operation supported on a vertex is zero on assignment components on which its active labels are not all at that vertex.

8.1 Advancing support with an arbitrary multiplier

Boundary-support methods also appear in tautological vanishing results such as [GV, Theorem 1.1], where support is measured by rational components. Here we lower the dimension of the image in the heavy base, using operational vanishing over smooth test families. We first make this induction precise. A term in it consists 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 over the vertices.

The operational hypothesis in this lemma is essential. In its applications below, it follows from identities over smooth test families, rather than from identities separately on geometric fibers. For a pure operation, Proposition 7.4 identifies its restriction with a pullback from a smooth curve power. Vanishing of its Chow class on that power is therefore operational vanishing and can be pulled further to any cycle multiplier. The formation of the projector and the projected diagonal commutes with these base changes: the relative diagonals and the integrations along smooth proper curve powers have the requisite flat or refined-Gysin base-change identities. For residual operations, the explicit base-divisor factors in Proposition 7.4 give the required open vanishing. The same is true of the controlled singleton relations below.

There is also no restriction on the multiplier hidden in our use of sign coinvariants. In characteristic zero, signed averaging identifies coinvariants with the corresponding signed invariant subspace. A factorization L*H of operations becomes a sum over shuffles of products of the separately averaged L and H, pulled to disjoint slot sets. In each summand, apply L first to the multiplier. The factor H remains an averaged operation on its own active slots; it is pulled equivariantly through later cuts before it acts. This proves that the lemma applies repeatedly, even after the multiplier has become an arbitrary cycle produced by an earlier step. More explicitly, every later factorization of H is first an equality of its signed-average representatives on its own active slots. Pull that equality to the full chart and only then cap with the multiplier. No permutation is applied to the multiplier, and no invariance of the multiplier is required. Restriction to an assignment component or pullback through a further cut preserves this equality.

Initially the heavy base has dimension 3g-3. Every advancement strictly lowers the base-image dimension. Thus on any branch of the finite expansions there can be at most

advancements before the support is empty.

8.2 The elementary vanishing tests

On a current chart, Proposition 7.4 expands an ancestor entry into pure entries at current vertices, products of a pair operation with a controlled singleton, and residual cubic operations. Call the pair-times-singleton terms noise entries. Perform these expansions one entry at a time. If a residual summand is chosen, apply Lemma 8.1 to it immediately, at the cost of that one entry. We therefore never retain residual entries while counting the pure and noise entries.

The pure entries at any one vertex always use the same chosen symmetric class G_{h,B}. This is important both for the finite coefficient bound and for the common-color test below.

8.3 A repeated-color test at a flagged vertex

The surface test of Lemma 6.4 is needed only while one vertex has almost all the genus. On every other chart a less expensive advancement follows from the flag-centered curve algebra.

8.4 The finite color budget

Set

These are integers. Reserve 4K+20 disjoint sets of m_0 basis colors each, and let S_0 be all the remaining colors. This is possible because

For a current graph put

Under refinement the maximum vertex genus cannot increase, so l cannot decrease. The number of edges cannot decrease either. Unmarked stability gives

An advancement by a residual entry, by a noise test, by a small-genus test, or by Lemma 8.3 will be called ordinary. These are descriptions of the cost in entries, not additional geometric assumptions.

The two regimes below use different information about the remaining colors. While l<K, we remove whole entries without changing the three colors of any retained entry; the reserved blocks can therefore be counted by their original triples. We use the repeated-color rewrites only once l\ge K. They can change which colors occur together in an entry, but preserve the total multiset of color occurrences before the advancing factor is removed. In that regime we count occurrences in S_0. Since l cannot decrease, we never need an original-triple count after making such a rewrite.

Small genus deficit: l<K. There is a unique vertex of genus b, since b>g-K>g/2. The total genus of all the others is at most l, and stability gives

For completeness, sum 2h_v-2+\operatorname{val}(v)\ge1 over all vertices except the large one. The sum equals 2l-\operatorname{val}(v_{\mathrm{large}}), proving the first inequality. The genus formula then gives the second.

After expanding the active entries, use a residual immediately. If a color occurs as a singleton in more than R_\Gamma noise entries, use Lemma 8.2. Otherwise there are at most g(10K+2) noise entries. If a small vertex has more than C_K pure entries, use the same lemma there. Otherwise the number of pure entries off the large vertex is at most 2K C_K. Each ordinary advancement in this regime costs at most

As specified above, retained entries in this regime still have their original triples, including those internal to each unused reserved block.

Suppose none of these ordinary tests applies. Choose a reserved color block E_0 that has not been used for a large-vertex test. Let D_0=\binom{m_0}{3}. Among its original D_0 entirely internal entries, the number that are no longer active pure entries at the large vertex is at most

Here (8.1) bounds all preceding ordinary losses. Earlier large-vertex tests used only entries internal to different, disjoint reserved blocks, so they removed none of these entries. The numerical inequalities follow from

Thus more than 0.99D_0 internal entries remain pure at the large vertex.

Consume the product of all those entries. On a smooth test base, obtain a section if necessary as in Lemma 8.1. Its dimension is at most 3g-2; since b>g-K,

We may therefore use Lemma 6.4. Choose an even integer e between m_0/2-2 and m_0/2. For d_e=\binom e3 and t_b=25(b-1)^2-1, the explicit bounds are

and consequently

For the first bound, m_0\ge128 and

The selected product thus vanishes modulo base support of codimension at least b-1. Applying Lemma 8.1, it either lowers the base-image dimension by at least b-2>g-K-2, or gives a boundary remainder requiring a strict refinement. Call this an expensive advancement and mark E_0 as used.

There are enough fresh blocks for every such advancement. Those that require a refinement start with fewer than 3K edges by (8.5); along a branch there are at most 3K+1 of them. Every other expensive advancement lowers the base-image dimension by at least g-K-2. There can be at most three such advancements with nonempty resulting support, because

A fourth can be terminal, making the support empty. In total at most 3K+5<4K+20 blocks are needed. No expensive advancement consumes an occurrence of a color in S_0.

Large genus deficit: l\ge K. Once this regime is reached it persists. We continue to expand ancestors using the same pure lift at each current vertex and peel residuals immediately. A noise test consumes at most 10g+1 entries. If no such test applies, at most 10g^2 noise entries remain.

All ordinary advancements on the entire preceding path, including those in the small-deficit regime, cost at most 11g entries each:

There have been at most 3g-2 advancements in total. Each consumed entry has three color occurrences. The cumulative ordinary loss of occurrences in S_0 is therefore less than 99g^2. The unpeeled noise accounts for at most another 30g^2. Expensive advancements used no S_0 colors. Hence the number of pure occurrences of colors from S_0 is at least

The allowance 140g^2 is larger than the sum 129g^2 of the two losses just obtained. The permutations used in Lemma 8.3 preserve the color multiset before its advancing factor is removed, so the same bound remains valid after any number of those rewrites.

Discard any identically zero product, in particular a product containing a pure entry with a repeated color. Suppose that every color i\in S_0, at every vertex v, belongs to at most \binom{h_v+1}{2} pure entries. The total number of pure occurrences would then be at most

This contradicts (8.8). Indeed, the lower bound minus this upper bound is

There is therefore a color and vertex to which Lemma 8.3 applies. Every vertex in this regime has a flag: a connected graph without any flags at one of its vertices could only be the one-vertex, no-edge graph, for which l=0. The lemma supplies another ordinary advancement, at the claimed cost.

8.5 Completion