Let k be an algebraically closed field of characteristic char(k) ≠ 2, 3, 5. For 0 ≤ n ≤ 12 we consider the moduli stack ℳ5, n of smooth genus-5 curves with n ordered, pairwise disjoint marked points. Our aim is to describe the rational Chow ring A*(ℳ5, n) and to strengthen this description by proving a generation property for Chow groups of products. The guiding principle is that genus 5 is the first genus in which the canonical embedding is uniformly controlled by low-degree equations on a large open locus, while still being sufficiently rigid to allow an explicit analysis of degenerations caused by marking points.
The central statements are twofold. First, we show that the Chow ring of ℳ5, n is tautological in the strongest expected sense: it is generated by the standard ψ-classes, κ-classes, and equivalently by ψ- and λ-classes. Second, we prove that ℳ5, n satisfies the Chow–K"unneth generation property (CKgP) introduced by Canning–Larson, which can be viewed as a robust compatibility of Chow with products against quotient-stratified stacks.
More precisely, for every 0 ≤ n ≤ 12 we prove:We emphasize that CKgP is strictly stronger than the tautologicality of A*(ℳ5, n); it is designed to be stable under constructions that occur naturally in inductive and stratified arguments (notably, passage to Grassmann bundles, projective bundles, and quotient stacks).
The integer 12 appears for a
geometric reason intrinsic to genus 5
canonical curves. On the nontrigonal locus, a genus-5 curve C is canonically embedded in ℙ4 and is a complete intersection
of three quadrics. The space of quadrics in ℙ4 has dimension
$$
h^0(\mathbb{P}^4,\mathcal{O}_{\mathbb{P}^4}(2))=\binom{4+2}{2}=15,
$$
and for a nontrigonal canonical genus-5
curve the ideal in degree 2 is
precisely a 3-dimensional subspace of
this 15-dimensional space. Thus the
curve is cut out by a , i.e. a 3-plane
in H0(ℙ4, 𝒪(2)).
This is the first reason that n = 12 is a natural endpoint for a uniform ``nets of quadrics’’ method.
A second, subtler reason is that n = 12 is also the first value of n at which the failure of independence is no longer a disjoint union of elementary degeneracy loci. For n = 8, 9 one can treat the non-independent locus by a single or a disjoint union of determinantal conditions. For n = 10, 11, while there are many determinantal components, they remain disjoint on the nontrigonal locus for geometric reasons (intersections force the curve into special gonality loci). At n = 12, distinct determinantal loci intersect on the nontrigonal locus, and these intersections reflect the presence of a g41 (tetragonal structure). Consequently, n = 12 is the first case requiring a refined overlap stratification governed by tetragonal geometry and the associated scroll in the canonical embedding. Our argument is built to handle precisely this new phenomenon.
The starting point for our approach is the method introduced by Liu for ℳ5, 8 (and extended in closely related forms to ℳ5, 9). In that range, one exploits that nontrigonal genus-5 curves are complete intersections of three quadrics, and one models the relevant moduli as an open substack of a Grassmann bundle of nets of quadrics over a configuration space of points in ℙ4. The key technical feature is that, on an open locus, the marked points impose independent conditions on quadrics; on that locus, the kernel of the evaluation map is a vector bundle and Grassmann bundle techniques give explicit generators of the Chow ring.
Our contribution is twofold. First, we uniformize the Grassmann-bundle presentation for all n ≤ 12 on the open locus of independence, giving a single framework that produces tautological generators and proves CKgP there. Second, and more importantly, we extend the method to the complement of the independence locus by giving an explicit stratification and a compatible set of geometric presentations for every stratum. The complement is governed by a concrete and computable condition: the presence of a canonical effective divisor supported on a subset of eight marked points. This reduces the failure of independence to a finite union of determinantal loci indexed by 8-subsets I ⊂ {1, …, n}, and it gives a precise combinatorial handle on all possible degenerations caused by marking points.
In particular, Liu’s analysis at n = 8 can be recovered as the case in which there is a single such 8-subset, and the corresponding degeneracy locus is described by stabilizing a hyperplane in the canonical ℙ4. For n = 9, one obtains a union of nine analogous components. The novelty from n = 10 onwards is not the determinantal nature of these components (which remains accessible by Thom–Porteous), but the need to manage their incidence and to present each locally closed stratum by a space with controllable Chow ring. This is especially pronounced at n = 12, where overlap strata reflect tetragonal pencils and require a further geometric encoding.
Canning–Larson introduced CKgP as a powerful structural property of moduli stacks and established it in a number of cases using stratifications, stability under basic operations, and control of Chow rings on special loci. Our results fit naturally into this framework. On the one hand, once a stack admits a stratification by locally closed substacks whose Chow rings are generated by explicitly described universal classes, tautologicality often follows by localization exact sequences. On the other hand, CKgP is designed to propagate through such arguments: it is stable under passing to vector bundles, projective bundles, Grassmann bundles, and certain quotient constructions, and it glues along stratifications under mild hypotheses.
We therefore organize the proof so that the same geometric input establishes both tautologicality and CKgP. Concretely, for each stratum we provide a presentation as an open substack of a Grassmann bundle over a configuration stack built from projective bundles over classifying stacks BG of linear algebraic groups G. Such building blocks are known to satisfy CKgP, and the open conditions are arranged so that the CKgP permanence results apply. This strategy is particularly well-suited to genus 5: the canonical model in ℙ4 brings in PGL5-equivariance, while passing to an SL5-lift introduces a universal rank-5 bundle with controlled Chern classes, allowing us to keep all Chow computations inside a manageable ring.
From the point of view of Chow rings, the main consequence is that the natural generators on the geometric models (Chern classes of universal bundles and subbundles, and hyperplane classes from projective bundles) can be identified with tautological classes on ℳ5, n. This identification uses standard tools: adjunction in the universal complete intersection, Grothendieck–Riemann–Roch to relate ambient bundles to the Hodge bundle f*ωf, and the basic relations among ψ-, κ-, and λ-classes. Once this is done, the Chow rings of the strata become tautological by construction, and CKgP follows from the same presentation.
We now summarize the proof strategy at a level sufficient to motivate the subsequent sections.
On the trigonal locus ℳ5, n3 and on the hyperelliptic locus ℳ5, n2, Chow groups are already understood to be tautological in the sense needed for gluing, by existing results on tautological support and by the structural results of Canning–Larson. We therefore work on the open complement ℳ5, n \ ℳ5, n3, where canonical curves are (2, 2, 2)-complete intersections in ℙ4. At the end, we glue back using localization sequences in Chow.
On the open locus 𝒰n where the marked points impose independent conditions on quadrics, the kernel of the quadratic evaluation map is a vector bundle En of rank 15 − n. Nets of quadrics are parametrized by G(3, En), and the open sublocus where the corresponding complete intersection is smooth gives a space Xn mapping to 𝒰n. This is the basic Grassmann presentation: it provides a uniform and explicit geometric source of Chow generators, and these generators can be identified with tautological classes on 𝒰n. In particular, A*(𝒰n) is tautological and satisfies CKgP.
The complement of 𝒰n is controlled by an
elementary but decisive lemma: failure of surjectivity of evaluation on
quadrics is equivalent to the existence of a nonzero section of ωC−1(Γ),
which in turn is equivalent to the existence of an effective canonical
divisor supported on a subset of eight of the marked points. Thus the
non-independent locus is a finite union ⋃|I| = 8DI,
where DI
is the locus where ∑i ∈ Ipi
is a canonical divisor. Each DI is a
determinantal locus defined by rank drop of the map
f*ωf → ⨁i ∈ Iσi*ωf,
so its fundamental class is given by a Thom–Porteous polynomial in λ- and ψ-classes. This yields
tautologicality of the classes of the boundary pieces entering
localization, and it identifies the combinatorics of the
stratification.
To complete the argument we must control the Chow groups of the strata themselves, not just their fundamental classes. For n ≤ 11, the geometry forces strong restrictions on how many distinct canonical 8-subsets can occur simultaneously on a nontrigonal curve with n marked points; in particular, for n = 10, 11 distinct DI are disjoint on the nontrigonal locus. Hence the stratification is relatively simple: the complement of 𝒰n is a disjoint union of strata each modeled on a single condition DI, and each stratum admits a Grassmann presentation obtained by stabilizing the hyperplane containing the canonical 8-tuple.
At n = 12 we must additionally analyze intersections DI ∩ DJ. These intersections are governed by the existence of degree-4 complements to canonical divisors, and on a genus-5 curve this forces a tetragonal pencil. The resulting strata are therefore described by configurations of fibers in a g41 (or, equivalently, by rulings on the associated scroll in ℙ4). We show that only finitely many overlap types occur, and for each type we construct an explicit configuration stack (built from quotients by stabilizers in PGL5 or its SL5-lift) and again a kernel bundle of an evaluation map on quadrics, whose Grassmannian of 3-planes contains an open substack isomorphic to the given stratum.
Once a stratum is presented as an open in a Grassmann bundle over a configuration stack, its Chow ring is generated by the Chern classes of the universal subbundle and by pullbacks of classes from the base. We verify that these generators correspond, under the moduli interpretation, to tautological classes (λ- and ψ-classes). Consequently A* of each stratum is tautological, and each stratum satisfies CKgP because it is built from CKgP pieces by operations preserving CKgP. Localization and stratified gluing then imply both tautologicality and CKgP for the whole ℳ5, n.
The restriction char(k) ≠ 2, 3, 5 is imposed to ensure that the canonical model of a nontrigonal genus-5 curve behaves as expected (in particular, as a smooth complete intersection of three quadrics in ℙ4), and to avoid pathologies in the intersection theory of the relevant quotient stacks and in the classical geometry of special linear series used in the overlap analysis at n = 12. Since we work with rational Chow groups, the resulting intersection-theoretic computations are insensitive to torsion phenomena but remain sensitive to the validity of the geometric classifications we invoke.
The next section recalls the geometry of genus-5 canonical curves on the nontrigonal locus and sets up the Grassmann-bundle parameter spaces of nets of quadrics. We also fix conventions for stacks versus coarse moduli spaces, and we recall the tautological ring and the formal properties of CKgP used throughout. Subsequent sections develop the independence criterion, the determinantal loci DI and their Thom–Porteous classes, and the Grassmann presentations of the strata needed for the localization arguments, with particular emphasis on the tetragonal overlap strata arising only when n = 12.
Let C be a smooth
genus-5 curve over k. If C is nonhyperelliptic, the canonical
linear system |ωC| is base
point free and very ample, hence defines the canonical embedding
φω: C ↪ ℙ(H0(C, ωC)∨) ≅ ℙ4.
We write V := H0(C, ωC),
so dim V = 5. The quadratic
equations of the canonical image are governed by the multiplication
map
By Riemann–Roch, deg (ωC⊗2) = 16
and h1(ωC⊗2) = 0,
hence
h0(C, ωC⊗2) = 16 − (5) + 1 = 12.
Since $\dim \mathrm{Sym}^2 V =
\binom{5+1}{2}=15$, surjectivity of is equivalent to dim ker (μC) = 3.
The kernel ker (μC) ⊂ Sym2V
is canonically identified with H0(ℙ4, ℐC(2)),
the space of quadrics containing the canonical curve.
For genus 5, the Petri theorem specializes to the following dichotomy: on the nontrigonal locus, the map μC is surjective, while on the trigonal locus it fails to be surjective by exactly the amount predicted by the presence of a g31. Concretely, if C is nontrigonal and nonhyperelliptic, then H0(ℙ4, ℐC(2)) is 3-dimensional, and the canonical model is scheme-theoretically cut out by quadrics. Moreover, since C ⊂ ℙ4 has codimension 3, the three independent quadrics containing it form a whose scheme-theoretic base locus is precisely C. In this case the canonical curve is a complete intersection of three quadrics, i.e. of type (2, 2, 2) in ℙ4.
We will systematically work on the open substack ℳ5 \ ℳ53, where this description holds uniformly. The characteristic assumption char(k) ≠ 2, 3, 5 ensures that the standard projective-geometric classification of canonical genus-5 curves (and the basic determinantal computations we use later) behaves as in characteristic zero; in particular, it excludes low-characteristic failures of Petri-type statements and of transversality for complete intersections of quadrics.
Fix a 5-dimensional k-vector space W. A net of quadrics in ℙ(W) ≅ ℙ4 is a 3-dimensional linear subspace N ⊂ Sym2W∨.
Nets are parametrized by the Grassmannian
G(3, Sym2W∨),
and we let S denote the
universal rank-3 subbundle on this
Grassmannian. Over ℙ(W) × G(3, Sym2W∨)
there is a tautological evaluation map
S → H0(ℙ(W), 𝒪ℙ(W)(2)) ⊗ 𝒪 → 𝒪ℙ(W)(2),
whose image defines a universal closed subscheme 𝒵 ⊂ ℙ(W) × G(3, Sym2W∨):
namely, 𝒵 is the common zero locus of
the three quadrics in the net. By construction, the fiber of 𝒵 → G(3, Sym2W∨)
over a point [N] is the
complete intersection ⋂Q ∈ NV(Q) ⊂ ℙ(W).
There is an open subset G(3, Sym2W∨)sm where this fiber is a smooth curve. Equivalently, writing the net as three quadratic equations Q1, Q2, Q3, smoothness is the condition that the Jacobian matrix (∂Qi/∂xj) have rank 3 along the intersection. This is an open condition: the singular locus is cut out by the 3 × 3 minors of the Jacobian together with the equations Qi. On G(3, Sym2W∨)sm the family 𝒵 is flat and smooth of relative dimension 1; by adjunction, each fiber has ω ≅ 𝒪(1) and genus 5. Thus G(3, Sym2W∨)sm provides a concrete parameter space for canonically embedded, nontrigonal genus-5 curves, before taking projective equivalence into account.
The group PGL(W) ≅ PGL5 acts
naturally on ℙ(W) and hence on
Sym2W∨,
preserving the open subset of smooth nets. Passing to the quotient by
this action accounts for the fact that the canonical embedding is unique
only up to projective automorphism of ℙ4. One obtains a morphism of
stacks
which is an isomorphism: given a nontrigonal curve C, its canonical model in ℙ4 is cut out by a 3-dimensional space of quadrics, and
conversely any smooth (2, 2, 2)-complete intersection in ℙ4 has genus 5 and is canonically embedded. We will use
(and its pointed analogues) to reduce Chow-theoretic questions about
ℳ5, n \ ℳ5, n3
to explicit computations on quotient stacks built from Grassmannians and
projective bundles.
Our arguments are naturally equivariant and are most efficiently formulated at the level of stacks. For a linear algebraic group G acting on a scheme (or algebraic space) X, we write [X/G] for the quotient stack. We will compute A*([X/G]) using the equivariant Chow theory of Edidin–Graham, in particular leveraging the fact that Chow rings of classifying stacks BG = [Spec(k)/G] and of projective/Grassmann bundles over them admit explicit descriptions in terms of Chern classes of universal bundles.
A technical issue in genus 5 is that
PGL5 has no tautological
rank-5 vector bundle on BPGL5. This becomes
inconvenient once we wish to write universal evaluation maps on ℙ4-bundles and describe their
Chern classes directly. We therefore pass to an SL5-lift. Concretely, we use the
central isogeny
1 → μ5 → SL5 → PGL5 → 1,
and we replace PGL5-quotients by SL5-quotients. On BSL5 there is a universal
rank-5 vector bundle V with c1(V) = 0, and
the universal ℙ4-bundle is
ℙV → BSL5.
The corresponding quotient stacks fit into a μ5-gerbe
[X/SL5] → [X/PGL5].
Since we work with rational coefficients, this change does not affect
Chow groups: the pullback along a finite banded gerbe is an isomorphism
on A*(−)ℚ. Thus,
for Chow-theoretic purposes, we may work systematically in the SL5-world, compute using the Chern
classes of V and its
associated bundles, and descend conclusions to the original PGL5-quotients (hence to moduli)
without loss.
In the pointed situation, the same philosophy applies. Configurations of marked points in the ambient ℙ4 will be encoded by iterated fiber products (ℙV)n over BSL5, and evaluation maps on quadrics will be expressed as morphisms of vector bundles built from Sym2V∨ and the line bundles ηi*𝒪ℙV(1). Although the detailed use of these evaluation maps belongs to the next section, the key point here is that the SL5-lift supplies the universal vector bundles needed to make these constructions canonical and to keep track of their Chow classes.
We recall the tautological classes used throughout. Let f: 𝒞5, n → ℳ5, n
be the universal curve with sections σ1, …, σn.
The cotangent line class at the i-th marking is
ψi := c1(σi*ωf) ∈ A1(ℳ5, n),
and the κ-classes are
κj := f*(c1(ωf)j + 1) ∈ Aj(ℳ5, n).
The Hodge bundle 𝔼 := f*ωf
has rank 5 on ℳ5, n, and we set
λj := cj(𝔼) ∈ Aj(ℳ5, n).
The tautological ring R*(ℳ5, n) ⊂ A*(ℳ5, n)
is the ℚ-subalgebra generated by the
ψ- and κ-classes; equivalently, one may use
ψ- and λ-classes, since κ-classes are expressible in terms
of Chern characters of f*ωf⊗m
by Grothendieck–Riemann–Roch, and λ-classes generate these Chern
characters in fixed genus.
When we pass to an open substack U ⊂ ℳ5, n, we keep the same notation for the restrictions of these classes to A*(U). In particular, on the nontrigonal open ℳ5, n \ ℳ5, n3, all tautological classes remain defined and will be compared with the natural generators coming from the Grassmann and projective bundle models discussed above.
We will repeatedly use the Chow–K"unneth generation property (CKgP)
of Canning–Larson. By definition, a finite-type stack Y has CKgP if for every
quotient-stratified finite-type stack X, the exterior product map
A*(Y) ⊗ A*(X) → A*(Y × X)
is surjective. The point of CKgP is not merely that A*(Y) is generated by
geometrically meaningful classes, but that these generators behave well
in products with stacks that arise in inductive arguments.
CKgP is designed to be stable under exactly these operations: it holds for BG, it is preserved under vector bundles, projective bundles, and Grassmann bundles, and it can be glued along finite stratifications by locally closed substacks satisfying mild hypotheses. Accordingly, once a stratum of ℳ5, n \ ℳ5, n3 is exhibited as an open substack of a Grassmann bundle over a base built from BG’s and projective bundles, CKgP for the stratum follows formally from these permanence properties. The same presentations also provide explicit Chow generators (Chern classes of universal bundles and hyperplane classes), which we will identify with tautological classes on the moduli side.
In summary, the background facts assembled in this section serve two purposes. First, they supply a uniform geometric model for nontrigonal genus-5 curves as smooth (2, 2, 2)-complete intersections in ℙ4, controlled by nets of quadrics. Second, they fix the stack-theoretic and intersection-theoretic framework—in particular the SL5-lift and the permanence of CKgP—that will allow us to convert projective geometry into tautological statements about Chow rings of ℳ5, n.
Fix 0 ≤ n ≤ 12 and consider
a pointed curve (C; p1, …, pn)
in the open nontrigonal locus ℳ5, n \ ℳ5, n3.
We write
Γ := p1 + ⋯ + pn
for the associated reduced effective divisor. In the canonical embedding
C ↪ ℙ4, the
relevant linear series of quadrics is H0(ℙ4, 𝒪(2)),
and the points Γ impose
conditions by evaluation at the marked points. The purpose of this
section is to isolate a concrete and essentially combinatorial
criterion, valid for n ≤ 12,
for when Γ fails to impose
independent conditions on quadrics. This criterion will be the input for
a finite stratification of ℳ5, n \ ℳ5, n3
by determinantal loci indexed by 8-subsets of {1, …, n}.
Let C ⊂ ℙ4 be
the canonical model of a nontrigonal genus-5 curve. We will say that Γ if the restriction map
is surjective. Since h0(ℙ4, 𝒪(2)) = 15,
surjectivity is equivalent to dim ker (evΓℙ4) = 15 − n,
and in families this is the condition that the kernel of the universal
evaluation map be a vector bundle of rank 15 − n.
Because Γ ⊂ C ⊂ ℙ4, the
map factors through the restriction to C:
On a canonically embedded curve we have 𝒪C(1) ≅ ωC,
hence 𝒪C(2) ≅ ωC⊗2.
Consequently, the second arrow in is simply evaluation of sections of
ωC⊗2
at the marked points:
evΓC: H0(C, ωC⊗2) → H0(Γ, ωC⊗2|Γ).
Thus, once we know that the first arrow in is surjective, the
independence condition on quadrics becomes a statement purely on the
curve C, i.e. a statement
about the vanishing of H1(C, ωC⊗2(−Γ))
via the standard exact sequence.
On the nontrigonal locus, C ⊂ ℙ4 is a smooth
complete intersection of three quadrics. In particular, its ideal sheaf
admits the Koszul resolution
0 → 𝒪ℙ4(−6) → 𝒪ℙ4(−4)⊕3 → 𝒪ℙ4(−2)⊕3 → ℐC → 0.
Twisting by 𝒪ℙ4(2) yields
Since H1(ℙ4, 𝒪(−4)) = H1(ℙ4, 𝒪(−2)) = 0,
the long exact sequence in cohomology applied to gives
H1(ℙ4, ℐC(2)) = 0.
Therefore the restriction map
is surjective. This is the form of ``Noether surjectivity’’ we require:
quadrics on the ambient ℙ4
generate all sections of ωC⊗2
on a smooth (2, 2, 2)-complete
intersection.
Combining with , we obtain:
Thus the failure of Γ to
impose independent conditions on quadrics is governed by a cohomology
group on C alone.
Consider the exact sequence on C
Taking global sections, we see that evΓC
is surjective if and only if
H1(C, ωC⊗2(−Γ)) = 0.
By Serre duality,
H1(C, ωC⊗2(−Γ)) ≅ H0(C, ωC−1(Γ))∨.
Hence Γ fails to impose
independent conditions on quadrics if and only if
We now convert into the existence of a canonical subdivisor supported on Γ. The key point is that any nonzero section of ωC−1(Γ) determines an injection of line bundles ωC ↪ 𝒪C(Γ), whose cokernel is supported on an effective divisor E ≤ Γ. Since deg ωC = 8 in genus 5, the complement Γ − E has degree 8 and is canonical. We record this in the form we will use later.
Combining , , and Lemma~, we obtain the promised criterion: on the nontrigonal locus, the only way n ≤ 12 marked points can fail to impose independent conditions on quadrics is by containing a canonical divisor of degree 8 among them.
We globalize the canonical subdivisor condition on the moduli stack.
Let I ⊂ {1, …, n} be
a subset of cardinality |I| = 8. Over ℳ5, n we have the Hodge
bundle 𝔼 = f*ωf
and the evaluation map at the marked points indexed by I,
At a geometric point (C; p), the fiber
of is the evaluation map
H0(C, ωC) → ⨁i ∈ IωC|pi.
Since dim H0(C, ωC) = 5,
this map drops rank if and only if there exists a nonzero canonical form
vanishing at all pi with i ∈ I, i.e. if and only if
∑i ∈ Ipi
is an effective canonical divisor. Equivalently, in the canonical
embedding C ↪ ℙ4,
the points {pi}i ∈ I
lie on a hyperplane section.
On the other hand, the independence condition on quadrics for $\Gamma=\sum_{i=1}^n p_i$ is a condition about surjectivity of . By the reductions above, we may reformulate this as the nonexistence of a canonical 8-subdivisor among the marked points, hence as the avoidance of all DI.
We emphasize two immediate consequences of this description.
First, for n ≤ 7 there are no 8-subsets, hence 𝒰n = ℳ5, n \ ℳ5, n3: any configuration of at most 7 marked points imposes independent conditions on quadrics on every nontrigonal genus-5 curve. Thus the first genuinely new case is n = 8, where the failure locus is a single divisor D{1, …, 8}, recovering the basic phenomenon that an 8-tuple may coincide with a canonical hyperplane section.
Second, Proposition~ provides a canonical finite indexing set for the boundary of 𝒰n, namely the collection of 8-subsets I ⊂ {1, …, n}. This finiteness is what makes the range n ≤ 12 tractable: while Γ may have many effective subdivisors, only degree 8 subdivisors can be canonical in genus 5, and there are only finitely many such subsets among n marked points.
For later use, it is convenient to encode the failure locus in terms of the set of canonical subsets realized by a pointed curve.
The assignment (C; p) ↦ 𝒮(C; p) yields a natural stratification of ℳ5, n \ ℳ5, n3 by locally closed substacks: we may first stratify by the cardinality |𝒮(C; p)|, and then refine by the incidence pattern among the subsets in 𝒮(C; p) (for instance, by the sizes of pairwise intersections I ∩ J, etc.). In practice, for n = 8, 9 this stratification is extremely simple, while for n = 10, 11 one can often show that distinct DI are disjoint on the nontrigonal locus, and for n = 12 one must allow nontrivial overlaps DI ∩ DJ governed by tetragonal geometry. We postpone these refinements, and the corresponding geometric descriptions of the strata, to later sections; at this stage we only require that the complement of 𝒰n is a union of the determinantal loci DI.
Finally, we note that the determinantal description will allow us to compute the classes [DI] by Thom–Porteous, hence to show that the boundary (ℳ5, n \ ℳ5, n3) \ 𝒰n is supported on tautological classes. This is the first input for the localization arguments used to glue Chow computations from 𝒰n to all of ℳ5, n \ ℳ5, n3, and ultimately to ℳ5, n itself.
We now give a uniform presentation of the independent locus 𝒰n ⊂ ℳ5, n \ ℳ5, n3 in terms of a Grassmann bundle of nets of quadrics, in a form convenient for Chow computations. The construction is a repackaging of Liu’s argument for n = 8, 9, but we emphasize that it works verbatim for all 0 ≤ n ≤ 12.
On ℳ5, n \ ℳ5, n3, every pointed curve (C; p1, …, pn) admits a canonical embedding C ↪ ℙ4 with ℙ4 = ℙ(H0(C, ωC)∨). Globally, the projective bundle of canonical hyperplanes is ℙ(𝔼∨), where 𝔼 = f*ωf is the Hodge bundle of rank 5. To work with honest vector bundles (rather than projective bundles) we pass to an SL5-lift.
Concretely, let $\widetilde{\mathcal{M}}_{5,n}\to
\mathcal{M}_{5,n}$ denote the μ5-gerbe of SL5-frames of 𝔼∨, i.e. the stack whose objects
over a scheme T are (C/T; σ1, …, σn)
together with an identification
$$
\phi\colon \mathcal{O}_T^{\oplus 5}\xrightarrow{\sim} \mathbb{E}^\vee
\qquad\text{with}\qquad \det(\phi)=1.
$$
Over $\widetilde{\mathcal{M}}_{5,n}$ we
obtain a rank-5 vector bundle
identified with the universal bundle V pulled back from BSL5, and hence a
universal projective space bundle ℙV → BSL5. The
key point for us is that rational Chow is insensitive to this μ5-banding: the
pullback
$$
\mathrm{A}^*(\mathcal{M}_{5,n})\longrightarrow
\mathrm{A}^*(\widetilde{\mathcal{M}}_{5,n})
$$
is an isomorphism (with ℚ-coefficients). Thus we may freely compute
on the SL5-cover and descend
at the end.
Let (ℙV)n be the
n-fold fiber product over
BSL5, with
projections ηi: (ℙV)n → ℙV.
We write
ξi := c1(ηi*𝒪ℙV(1)) ∈ A1((ℙV)n)
for the i-th hyperplane class.
The Chow ring of (ℙV)n is
generated over A*(BSL5) = ℚ[c2(V), c3(V), c4(V), c5(V)]
by the ξi,
with the usual projective bundle relations.
Consider the universal quadratic evaluation morphism on (ℙV)n:
whose fiber at (x1, …, xn) ∈ (ℙ4)n
is evaluation of quadrics at the points xi. Let
Vn ⊂ (ℙV)n
be the maximal open substack where evn is surjective. On
Vn the
kernel
$$
0\longrightarrow E_n \longrightarrow \mathrm{Sym}^2 V^\vee\big|_{V_n}
\xrightarrow{\ \mathrm{ev}_n\ } \bigoplus_{i=1}^n
\eta_i^*\mathcal{O}_{\mathbb{P}V}(2)\big|_{V_n}
\longrightarrow 0
$$
is a vector bundle of rank 15 − n. Since n ≤ 12, we have rk(En) ≥ 3, so
it makes sense to consider the Grassmann bundle of 3-planes in En.
We set
π: G(3, En) → Vn
and denote by S ⊂ π*En
the universal rank-3 subbundle. A point
of G(3, En)
therefore consists of a configuration of n points (x1, …, xn) ∈ Vn
together with a 3-dimensional subspace
W ⊂ H0(ℙ4, 𝒪(2))
of quadrics vanishing at all xi.
Smoothness and disjointness are open conditions, so Xn is indeed open. Equivalently, Xn is obtained by removing the Jacobian-minor locus (where the three quadrics fail to meet transversely) and the incidence diagonals from G(3, En).
By construction, Xn carries a
universal family
𝒞n ⊂ ℙV×BSL5Xn
cut out by the universal quadrics corresponding to S ⊂ π*En ⊂ π*Sym2V∨.
Writing gn: 𝒞n → Xn
for the projection and 𝒪𝒞n(1) for the
restriction of 𝒪ℙV(1), adjunction for a
(2, 2, 2)-complete intersection
gives
Moreover, the n marked points
define n sections σ̃i: Xn → 𝒞n
induced by the tautological points (x1, …, xn);
these exist precisely because we built Xn as a locus
where the xi lie on the
curve.
There is a natural SL5-action on (ℙV)n, hence on
Vn, En, G(3, En), and
Xn.
Passing to quotient stacks, the family (𝒞n → Xn; σ̃1, …, σ̃n)
descends to an n-pointed
family of smooth genus-5 curves over
[Xn/SL5],
and thus defines a morphism
The image of is exactly 𝒰n. Indeed, by definition
of Vn, the
marked points impose independent conditions on quadrics in ℙ4, and by the embedding is
canonical. Conversely, given (C; p1, …, pn) ∈ 𝒰n,
the canonical model of C is a
smooth (2, 2, 2)-intersection cut out
by a 3-dimensional space IC(2) ⊂ H0(ℙ4, 𝒪(2)).
Since the points impose independent conditions on quadrics, IC(2) maps into
the fiber of En at (p1, …, pn),
producing a point of G(3, En) lying
in Xn.
After choosing an SL5-frame
of H0(C, ωC)∨
(i.e. passing to the μ5-gerbe), this
construction is functorial, yielding an inverse to over 𝒰n. In particular we
obtain an identification
where $\widetilde{\mathcal{U}}_n\to
\mathcal{U}_n$ is the induced μ5-gerbe. We will work
with [Xn/SL5]
in Chow.
The presentation reduces Chow computations on 𝒰n to standard computations on projective bundles and Grassmann bundles, provided we can identify the resulting generators with tautological classes.
On Xn, the Chow ring A*(Xn) is generated by:Indeed, G(3, En) → Vn is a Grassmann bundle and Xn ⊂ G(3, En) is open, so A*(Xn) is generated by the restrictions of the usual Grassmann generators. Moreover, the exact sequence defining En shows that c(En), hence c(S), is expressible in terms of c(Sym2V∨) and the Chern classes of ηi*𝒪ℙV(2), i.e. in terms of ci(V) and the ξi. Thus no new classes arise from S beyond these ambient generators.
We now relate these classes to λ- and ψ-classes on 𝒰n. Let (𝒞n → Xn; σ̃i)
be the universal curve as above. By , we have ωgn ≅ 𝒪𝒞n(1),
hence
σ̃i*ωgn ≅ σ̃i*𝒪𝒞n(1) ≅ ηi*𝒪ℙV(1)|Xn.
Therefore
For the Hodge classes, we apply Grothendieck–Riemann–Roch to gn and ωgn.
Since ωgn ≅ 𝒪𝒞n(1)
and 𝒞n ⊂ ℙV × Xn
is cut out by three relative quadrics, the standard Koszul resolution
shows that gn*ωgn
is identified with V∨ pulled back from BSL5. In
particular,
and therefore the Chern classes of the Hodge bundle pull back to
polynomials in the ci(V).
Equivalently, on $\widetilde{\mathcal{U}}_n$ we have
Combining and , we conclude that the natural generators of A*(Xn) coming from the SL5-linear algebra model map to tautological generators on $\widetilde{\mathcal{U}}_n$, and hence on 𝒰n itself. Since $\mathrm{A}^*(\mathcal{U}_n)\cong \mathrm{A}^*(\widetilde{\mathcal{U}}_n)$ rationally, this yields the desired uniform description: the Chow ring of the independent locus is generated by ψ- and λ-classes, and all further classes obtained from the Grassmann presentation (such as c(S)) are polynomials in these by the exact sequence defining En.
Finally, we record the geometric endpoint at n = 12. In this case rk(E12) = 3, so G(3, E12) → V12 is a point-bundle and X12 ⊂ V12 is simply the open where the unique net of quadrics through the 12 points cuts out a smooth complete intersection. This is the precise sense in which n = 12 is the largest value for which the independent-locus Grassmann model remains available.
Fix an integer n with 0 ≤ n ≤ 12, and work throughout on
the open substack
ℳ5, n \ ℳ5, n3,
so that every curve is canonically embedded as a smooth complete
intersection of three quadrics in ℙ4. The purpose of this section is
twofold. First, we isolate the natural closed loci on which the marked
points fail to impose independent conditions on quadrics, and we compute
their fundamental classes by determinantal methods. Second, we explain a
uniform linear-algebraic presentation of these loci in terms of
configuration stacks attached to hyperplane stabilizers together with
Grassmann/affine-bundle constructions. This presentation is the input
for the excision arguments in the subsequent cases n = 10, 11 (and, with additional
refinements, for n = 12).
Let I ⊂ {1, …, n}
be an 8-element subset. Consider the
universal curve
f: 𝒞5, n → ℳ5, n
with sections σi, and the
natural evaluation morphism
We define DI ⊂ ℳ5, n \ ℳ5, n3
to be the (scheme-theoretic) degeneracy locus where evI drops rank,
i.e. where rk(evI) ≤ 4.
Equivalently, DI is the locus
of pointed curves (C; p1, …, pn)
for which there exists a nonzero canonical section s ∈ H0(C, ωC)
vanishing at the eight marked points {pi}i ∈ I.
Since deg (ωC) = 8,
such a section has divisor of zeros precisely ∑i ∈ Ipi,
and hence
𝒪C(∑i ∈ Ipi) ≅ ωC.
On the canonical model C ↪ ℙ(H0(C, ωC)∨) ≅ ℙ4,
sections of ωC are
identified with hyperplanes in ℙ4, and the zero divisor of s is the hyperplane section. Thus,
for (C; p) ∈ ℳ5, n \ ℳ5, n3,
the condition (C; p) ∈ DI
is equivalent to the existence of a hyperplane H ⊂ ℙ4 such that
C ∩ H = ∑i ∈ Ipi as
effective divisors on C.
On the open where the eight points span a unique ℙ3 ⊂ ℙ4, this
hyperplane is uniquely determined. This uniqueness is the basic reason
that hyperplane stabilizers (rather than all of PGL5) govern the geometry of DI.
We emphasize that, by construction, each DI is closed in ℳ5, n \ ℳ5, n3. Later, using Lemma~3.1 (the ``canonical subdivisor’’ criterion), we will identify the complement of the independent locus 𝒰n with the union ⋃|I| = 8DI. For the present section, we only use that the DI provide canonical closed pieces of the non-independent locus, with classes that are directly computable.
We now compute the fundamental class of DI in the Chow
ring of ℳ5, n \ ℳ5, n3.
The bundles in have ranks
rk(f*ωf) = 5, rk(⨁i ∈ Iσi*ωf) = 8,
and the degeneracy condition is rk(evI) ≤ 4. The expected
codimension is therefore
(5 − 4)(8 − 4) = 4.
On our nontrigonal locus, this expected codimension holds (in
particular, DI has no
components of smaller codimension), and Thom–Porteous applies without
correction terms.
For later use it is convenient to record an explicit expansion. Write
ej(I)
for the degree-j elementary
symmetric polynomial in the {ψi}i ∈ I.
Expanding c(𝔼)−1 up
to degree 4 yields
c(𝔼)−1 = 1 − λ1 + (λ12 − λ2) − (λ13 − 2λ1λ2 + λ3) + (λ14 − 3λ12λ2 + λ22 + 2λ1λ3 − λ4) + ⋯.
Therefore
No further geometry is used in this computation: it is purely
determinantal, and it already shows that the natural closed pieces of
the non-independent locus have tautological fundamental classes.
While Proposition~ controls [DI], for excision and CKgP we also require an explicit presentation of DI by quotient stacks built from projective and Grassmann bundles. We now describe such a presentation on a dense open substack of DI; this is sufficient for Chow computations because we will work by stratifying DI further by simple incidence conditions (distinctness, spanning, transversality), each defined by removing closed subsets.
We pass to the SL5-lift
as in Section~, so that we may work over classifying stacks with
universal vector bundles. Fix a reference hyperplane H0 ⊂ ℙ4, and
let
G ⊂ SL5
be the stabilizer of H0. Over BG there is a tautological
short exact sequence of vector bundles
with rk(U) = 4, rk(V) = 5, and det (V) ≅ 𝒪; geometrically ℙU ⊂ ℙV is the universal
hyperplane fixed by the choice of G.
We now isolate the data intrinsic to a canonical hyperplane section.
Choose and fix an ordering I = {i1, …, i8}
and single out i8
(this is purely notational and produces a cover of DI by isomorphic
pieces). Consider the configuration stack
𝒱I := (ℙU)7
over BG, with factors
indexed by i1, …, i7.
Let ζa
denote the hyperplane class pulled back from the a-th factor. There is a universal
evaluation map on quadrics in the hyperplane:
Let 𝒱I∘ ⊂ 𝒱I
be the open locus where ev7H is
surjective and the seven points are pairwise distinct. On 𝒱I∘, the
kernel
$$
0\longrightarrow K_I \longrightarrow \mathrm{Sym}^2
U^\vee\big|_{\mathcal{V}_I^\circ}
\xrightarrow{\ \mathrm{ev}^{H}_{7}\ } \bigoplus_{a=1}^{7}
\mathcal{O}(2\zeta_a)\big|_{\mathcal{V}_I^\circ}
\longrightarrow 0
$$
is a rank-3 vector bundle. Fiberwise,
KI is the
net of quadrics in H0 ≅ ℙ3
vanishing at the chosen seven points. For a general configuration, the
base locus of this net is a reduced scheme of length 8; we enforce this by restricting further to
an open substack 𝒱ICay ⊂ 𝒱I∘
where the common zero locus of KI in ℙU is finite étale of degree 8. Over 𝒱ICay the
residual base point defines an additional section, hence an induced
eighth point in ℙU; we view
this as providing the marked point pi8
in the hyperplane.
We now incorporate the remaining marked points (those not in I) as points of ℙV. Set
𝒲I, n := 𝒱ICay×BG(ℙV)n − 8,
where the factors of (ℙV)n − 8 are
indexed by {1, …, n} \ I. We impose
the open condition that these additional points are pairwise distinct
and disjoint from the eight points in ℙU; we continue to denote the
resulting open by 𝒲I, n.
The last ingredient is to lift the hyperplane net KI ⊂ Sym2U∨
to a net of quadrics in ℙV.
Using , we have a G-equivariant decomposition
The projection Sym2V∨ → Sym2U∨
is restriction of quadrics from ℙV to ℙU, and its kernel
N := (U∨ ⊗ L∨) ⊕ Sym2L∨
is the 5-dimensional space of quadrics
vanishing identically on ℙU
(i.e. the ideal generated by the hyperplane equation). A lift of the net
KI is
therefore equivalent to choosing a rank-3 subspace W ⊂ Sym2V∨
mapping isomorphically onto KI, or
equivalently to choosing a G-equivariant splitting of KI into Sym2V∨, i.e. a
morphism KI → N.
This is parametrized by the total space of the vector bundle
$$
\mathcal{H}_{I}\ :=\ \underline{\mathrm{Hom}}(K_I,N)\ \cong\
K_I^\vee\otimes N
$$
of rank 15 on 𝒲I, n.
Finally, we impose the condition that the lifted net vanishes at the
n − 8 marked points in ℙV. Fiberwise, this is an
affine-linear condition on $\underline{\mathrm{Hom}}(K_I,N)$; after
choosing the graph description of the lift using , it becomes a linear
condition on the parameter KI → N.
Concretely, for each additional marked point x ∈ ℙV, evaluation of
elements of N at x gives a linear functional N → k, hence a linear
map
KI∨ ⊗ N → KI∨.
Taking the direct sum over the n − 8 points yields a morphism of
vector bundles
and we define ℰI, n := ker (ΘI, n)
on the maximal open locus where ΘI, n
is surjective. Since rk(KI∨) = 3,
the expected rank is
rk(ℰI, n) = 15 − 3(n − 8) = 39 − 3n,
which is ≥ 3 precisely for n ≤ 12. This numerology mirrors the
independent-locus picture: the value n = 12 is again the endpoint at
which the lift becomes essentially rigid.
Let 𝒳I, n ⊂ Tot(ℰI, n)
be the open locus where the corresponding three lifted quadrics cut out
a smooth complete intersection curve C ⊂ ℙV, and where all n marked points lie on C and are pairwise disjoint. The
construction is G-equivariant,
and we obtain a morphism
[𝒳I, n/G] → ℳ5, n \ ℳ5, n3.
Lemma~ shows that DI admits a covering by quotient stacks [𝒳I, n/G] where 𝒳I, n is an open substack of the total space of a vector bundle over a configuration stack built from (ℙU)7 and (ℙV)n − 8. In particular, 𝒳I, n is obtained from BG by iterating the operations of forming projective bundles, taking kernels of surjective maps of vector bundles on suitable opens, and removing determinantal closed substacks (Jacobian-minor and diagonal loci). This is the structural input that will allow us to deduce both tautologicality and CKgP for the pieces appearing in excision.
On the level of generators, the Chow rings of the ambient stacks are generated by Chern classes of the universal bundles U, V, L on BG together with the hyperplane classes from the various ℙU and ℙV factors, and (if one chooses to repackage Tot(ℰI, n) as a Grassmann bundle) by the Chern classes of the corresponding universal subbundles. Under the moduli map, these classes identify with tautological classes on DI: hyperplane classes pulled back along the marked sections agree with ψ-classes by the canonical adjunction ωC ≅ 𝒪C(1) on a (2, 2, 2)-intersection, and the Chern classes of V identify with λ-classes via the description of the Hodge bundle on the SL5-lift. We will make these identifications systematic when proving tautological generation on each stratum.
In the next section we exploit a key simplification for n = 10, 11: on the nontrigonal locus, the components DI are pairwise disjoint. This turns localization sequences in Chow into direct-sum decompositions over the DI, and the presentation above provides the explicit geometric model needed to compute A*(DI) and to verify that its generators descend to tautological classes on ℳ5, n.
We continue to work on the open substack
ℳ5, nnt := ℳ5, n \ ℳ5, n3,
so that every fiber is canonically embedded as a smooth complete
intersection of three quadrics in ℙ4.
For n ≤ 12, let 𝒰n ⊂ ℳ5, nnt
denote the independent locus for quadrics, and recall that the natural
closed pieces of the complement are the degeneracy loci DI indexed by
|I| = 8.
For n = 10, 11 a key simplification occurs: on ℳ5, nnt the substacks DI are pairwise disjoint. Consequently, localization sequences in Chow do not involve any higher overlap strata, and the global argument reduces to understanding 𝒰n and each individual component DI.
Fix n ∈ {10, 11} and two
distinct 8-subsets I, J ⊂ {1, …, n}.
Write
Γ := p1 + ⋯ + pn, A := Γ − ∑i ∈ Ipi, B := Γ − ∑j ∈ Jpj,
so deg (A) = deg (B) = n − 8,
i.e. deg (A) = 2 for n = 10 and deg (A) = 3 for n = 11.
In particular, for n = 10, 11 the non-independent locus
inside ℳ5, nnt is a
disjoint union of the DI. We denote
this closed substack by
𝒵n := ℳ5, nnt \ 𝒰n = ⨆|I| = 8DI,
using the canonical-subdivisor criterion (Lemma~3.1) to identify the
complement of 𝒰n
with ⋃|I| = 8DI,
and Corollary~ to upgrade the union to a disjoint union.
For n = 10, 11 the relevant
ranks are
rk(ℰI, 10) = 39 − 30 = 9, rk(ℰI, 11) = 39 − 33 = 6,
so in both cases ℰI, n has rank
≥ 3, and the resulting lifted nets of
quadrics vary in positive-dimensional families.
We record the consequence in the form we will use for Chow-theoretic arguments.
For n = 10, 11 we emphasize that Proposition~ is applied only componentwise on a single DI; no compatibility across different I is required, precisely because the DI are disjoint.
We now deduce tautological generation of Chow on each DI, and we note that the same argument supplies CKgP for DI.
We now combine the independent-locus computation with the disjointness and componentwise control of the DI.
Let j: 𝒰n ↪ ℳ5, nnt
be the open immersion and i: 𝒵n ↪ ℳ5, nnt
its closed complement. Localization gives an exact sequence
By Corollary~ we have 𝒵n = ⨆|I| = 8DI,
hence
A*(𝒵n) ≅ ⨁|I| = 8A*(DI).
On the other hand, 𝒰n is treated by the Grassmann-bundle model for nets of quadrics: on 𝒰n the kernel of quadratic evaluation is a vector bundle En of rank 15 − n, and the universal net is a 3-plane in En. Thus 𝒰n is an open substack of the Grassmann bundle G(3, En) over the evaluation-surjective locus Vn ⊂ (ℙV)n, and its Chow ring is generated by the standard Grassmann/projective generators, which identify with λ- and ψ-classes on 𝒰n. In particular, A*(𝒰n) is tautological.
For the closed part, Proposition~ gives tautological generation of A*(DI) for every I. Since pushforward along a regular closed immersion is compatible with multiplication by the fundamental class, and since [DI] itself is tautological by Thom–Porteous (Proposition~), the image i*(A*(DI)) ⊂ A*(ℳ5, nnt) lies in the tautological subring. Therefore, the exact sequence shows that A*(ℳ5, nnt) is generated by tautological classes.
Finally, we glue back across the trigonal locus. By the results of Canning–Larson on tautologicality (and CKgP) for the relevant low-gonality strata and their complements, and by the same localization argument applied to the open immersion ℳ5, nnt ↪ ℳ5, n, we conclude:
The argument isolates precisely what fails at n = 12: disjointness of the DI. When n = 12, distinct canonical 8-subdivisors can coexist without forcing gonality ≤ 3, and the resulting overlap strata must be analyzed using tetragonal geometry, to which we turn next.
We continue to work on
ℳ5, 12nt := ℳ5, 12 \ ℳ5, 123,
and we write Γ := p1 + ⋯ + p12.
For |I| = 8 we retain the
notation DI ⊂ ℳ5, 12nt
for the canonical-subdivisor locus. The new feature at n = 12 is that distinct DI can meet
without forcing gonality ≤ 3; thus the
localization argument for ℳ5, 12nt requires a
refinement of the stratification by iterated intersections of the DI.
Fix two distinct 8-subsets I ≠ J ⊂ {1, …, 12}. As
before we set
A := Γ − ∑i ∈ Ipi, B := Γ − ∑j ∈ Jpj,
so deg (A) = deg (B) = 4. The
defining property (C; p) ∈ DI
(resp. DJ)
is
𝒪C(∑i ∈ Ipi) ≅ ωC (resp.
𝒪C(∑j ∈ Jpj) ≅ ωC),
hence ∑i ∈ Ipi ∼ ∑j ∈ Jpj,
and subtracting from Γ gives
A ∼ B.
Lemma~ is the precise replacement, at n = 12, for the disjointness used at
n = 10, 11: overlaps do occur,
but they are forced into tetragonal geometry. We emphasize that the
pencil is intrinsically determined by Γ and ωC:
L ≅ 𝒪C(Γ) ⊗ ωC−1,
since A = Γ − (canonical
divisor). Consequently, for a fixed pointed curve (C; p), every 8-subset I with (C; p) ∈ DI
has the property that the complementary 4-subset Ic is a fiber of
the unique base-point-free pencil |L| produced above.
We now translate the existence of the g41 into
linear-algebraic data in ℙ4.
Let C ↪ ℙ4 be the
canonical embedding, so ωC ≅ 𝒪C(1).
If D ∈ |L| is a fiber
of the g41, then
deg (D) = 4 and h0(ωC(−D)) = h0(D) = 2,
hence D spans a plane ⟨D⟩ ≅ ℙ2 ⊂ ℙ4.
As D varies in the pencil
|L| ≅ ℙ1, the union
of these planes is a rational scroll threefold
ΣL := ⋃D ∈ |L|⟨D⟩ ⊂ ℙ4,
and ΣL is
a quadric hypersurface of rank 4 (a
cone over a smooth quadric surface). One convenient way to see this,
which is well adapted to our later quotient presentations, is to note
that ΣL is
the image of ℙ(φ|L|*ωC)
under the tautological map to ℙ4, and for g = 5, d = 4 this scroll has degree 2, hence is cut out by a single quadric.
In these terms, the overlap condition (C; p) ∈ DI ∩ DJ
acquires the following geometric meaning. The complements A = Ic
and B = Jc
are fibers of |L|, hence each
determines a ruling plane ΠA := ⟨A⟩
and ΠB := ⟨B⟩
on ΣL. The
two associated canonical divisors ∑i ∈ Ipi
and ∑j ∈ Jpj
correspond to hyperplane sections HI ∩ C
and HJ ∩ C.
Since |I ∩ J| = 4 by
Lemma~, the four common marked points lie on both hyperplanes; thus they
lie on the plane
ΛI, J := HI ∩ HJ ≅ ℙ2.
Conversely, the pencil of hyperplanes through ΛI, J,
ℙ1 ≅ { H ⊂ ℙ4 ∣ ΛI, J ⊂ H },
cuts out on C the pencil |L|, with moving part the residual
degree-4 divisor to ΛI, J ∩ C.
In particular, the tetragonal pencil extracted from DI ∩ DJ
can be recovered directly from the pair of hyperplanes (HI, HJ)
without introducing L a
priori.
For (C; p) ∈ ℳ5, 12nt, let 𝒮(C; p) be the set of 8-subsets I such that (C; p) ∈ DI. By the criterion of canonical 8-subdivisors, 𝒮(C; p) ≠ ∅ if and only if (C; p) ∉ 𝒰12. The refined stratification we require is obtained by fixing 𝒮(C; p) as a combinatorial datum, and then further separating according to whether the associated pencil |L| has 2 or 3 fibers supported on the marked points.
More concretely, suppose 𝒮(C; p) contains at least two elements I ≠ J. Then the complements Ic and Jc are two distinct fibers of the base-point-free pencil |L|. Because different fibers are disjoint and have degree 4, there can be at most three fibers of |L| contained in the 12 marked points. Accordingly, there are two basic regimes:In the ternary case, the overlap is genuinely higher: each pair of DI meets, and the triple intersection D Γ − F1 ∩ D Γ − F2 ∩ D Γ − F3 is nonempty and contains precisely those configurations for which Γ is a disjoint union of three fibers of a tetragonal pencil. We record the finiteness of possible patterns as follows.
Lemma~ provides a workable stratification of ℳ5, 12nt \ 𝒰12 by locally closed substacks indexed by overlap type. In the sequel we denote by 𝒵12(1) the locus where |𝒮(C; p)| = 1, by 𝒵12(2) the binary overlap locus |𝒮(C; p)| = 2, and by 𝒵12(3) the ternary overlap locus |𝒮(C; p)| = 3. The key point is that 𝒵12(2) and 𝒵12(3) admit explicit presentations reflecting the tetragonal scroll ΣL.
We explain how to construct quotient models for the overlap strata in a form suitable for Chow-theoretic localization. The guiding principle is that, on 𝒵12(2) and 𝒵12(3), the data of two (or three) canonical 8-subdivisors canonically produces extra linear subspaces in ℙ4 (the planes ΛI, J and the ruling planes ΠIc), and we pass to an SL5-lift in which we rigidify these subspaces by choosing a reference flag.
For the binary overlap locus, fix an ordered pair (I, J) with I ≠ J and consider the
locally closed stratum
𝒵I, J(2) := (DI ∩ DJ) \ ⋃K ∉ {I, J}DK,
so that 𝒮(C; p) = {I, J}
on 𝒵I, J(2).
On this stratum the associated hyperplanes HI and HJ are distinct
and uniquely determined (each is the unique hyperplane cutting out the
corresponding effective canonical divisor), and Λ := HI ∩ HJ
is a plane. We rigidify by choosing a reference flag of vector
subspaces
W0 ⊂ U0 ⊂ V0 ≅ k5, dim W0 = 3, dim U0 = 4,
so that ℙW0 ⊂ ℙU0 ⊂ ℙV0
models Λ ⊂ HI ⊂ ℙ4.
Let G(3, 4) ⊂ SL5
be the stabilizer of this flag; then on BG(3, 4) we have
universal bundles W ⊂ U ⊂ V of ranks
3, 4, 5 with c1(V) = 0.
Imposing only open conditions (pairwise distinctness of all 12 points and avoidance of the natural diagonals) yields an open substack 𝒱I, J∘ ⊂ 𝒱I, J.
On 𝒱I, J∘
we consider the quadratic evaluation map
evI, J: Sym2V∨ → ⨁m = 112𝒪(2)|pm,
where the targets are the pullbacks of 𝒪ℙV(2) along the
universal point morphisms. After restricting to the open locus where
evI, J is
surjective, its kernel defines a vector bundle EI, J.
The moduli of complete intersections of three quadrics through the 12 marked points is then obtained by choosing
a 3-plane in EI, J,
i.e. by passing to the Grassmann bundle G(3, EI, J).
Finally, removing the closed locus where the corresponding three
quadrics fail to intersect transversely (and removing the loci where the
resulting curve does not contain the points with the prescribed
hyperplane-section incidence) defines an open substack 𝒳I, J(2) ⊂ G(3, EI, J).
By construction there is a natural morphism
[𝒳I, J(2)/G(3, 4)] → ℳ5, 12nt,
and one checks that its image is precisely 𝒵I, J(2),
with representable and generically finite fibers; shrinking 𝒳I, J(2)
further by an open condition (excluding loci with extra canonical 8-subdivisors) yields an isomorphism onto
𝒵I, J(2).
The ternary overlap locus is treated similarly, but with one
additional fiber condition. Fix a partition {1, …, 12} = F1 ⊔ F2 ⊔ F3
into three 4-subsets and define the
locally closed stratum
𝒵F•(3) := D Γ − F1 ∩ D Γ − F2 ∩ D Γ − F3 \ ⋃K ∉ {Γ − F1, Γ − F2, Γ − F3}DK.
On 𝒵F•(3)
the line bundle L ≅ 𝒪C(F1) ≅ 𝒪C(F2) ≅ 𝒪C(F3)
is a base-point-free g41, and the
three divisors Fi are three
distinct fibers. Geometrically, this determines the quadric scroll ΣL ⊂ ℙ4
together with three ruling planes Πi = ⟨Fi⟩.
We rigidify by fixing a reference rank-4 quadric Q0 ⊂ ℙ4 and a
choice of ruling, and we let GQ ⊂ SL5
be the stabilizer of (Q0, ruling). Over BGQ we
then build a configuration stack parameterizing three planes in the
fixed ruling (i.e. three points of the ℙ1 parameter space of the ruling)
together with 4 ordered points on each
plane, again with diagonals removed. As in the binary case, surjectivity
of evaluation on quadrics gives a kernel bundle EF•,
and an open substack of G(3, EF•)
cut out by transversality yields a quotient presentation of 𝒵F•(3).
We summarize the outcome in the form needed later.
The constructions above reduce the n = 12 overlap analysis to explicit linear-algebraic parameter spaces. In the next section we will identify the resulting natural Chow generators (Chern classes from the stabilizer-universal bundles and hyperplane classes from the point factors, together with the Grassmann-subbundle Chern classes) with λ- and ψ-classes on the corresponding moduli strata, which is the input required to conclude tautologicality and CKgP via localization.
We show that all of these generators identify with polynomials in
λ- and
ψ-classes on the corresponding
moduli stratum. The resulting statement is uniform; the
only place where an extensive case distinction is required is in the of
the
finite list of strata in the n = 12 overlap locus
(cf. Lemma~
and Proposition~). Once a stratum has been
presented in the Grassmann form, the tautological identification is
formal.
Let 𝒵 ⊂ ℳ5, nnt be
any stratum for n ≤ 12
that
admits an SL5-lift and a
quotient presentation as in Theorem~D and, in the overlap
case n = 12, as in
Proposition~. Thus we have
𝒵 ≅ [𝒳/G],
where 𝒳 ⊂ G(3, E) is an open
substack of a Grassmann bundle over a
configuration stack 𝒱 obtained from
BG by iterating
projective bundles and
imposing only open conditions. Write π: 𝒳 → 𝒱 for the
projection.
The SL5-lift provides a
rank-5 vector bundle V on BG with
c1(V) = 0,
and hence on 𝒱 and 𝒳. Over 𝒱 we
also
have tautological morphisms to ℙV (or to ℙW, ℙU, etc., in
the overlap strata), corresponding to the marked points. For each marked
point we obtain a
line bundle pi*𝒪ℙV(1)
on 𝒱, and we denote
hi := c1(pi*𝒪ℙV(1)) ∈ A1(𝒱),
and by the same symbol its pullback to 𝒳 and 𝒵.
To compare hi with the
class ψi
on 𝒵, we recall the standard
feature
of the SL5-lift: the
projective bundle of canonical spaces is unchanged, but the
tautological 𝒪(1) differs by a twist.
More precisely, let 𝔼 := f*ωf
be the Hodge bundle on 𝒵, and let $\widetilde{\mathcal{Z}}\to \mathcal{Z}$ be
the
μ5-gerbe on which a
fifth root ℒ of det (𝔼) is chosen. Then
𝔼 ≅ V ⊗ ℒ on $\widetilde{\mathcal{Z}}$, and
ℙ(𝔼∨) ≅ ℙ(V∨)
with
𝒪ℙ(𝔼∨)(1) ≅ 𝒪ℙ(V∨)(1) ⊗ π*ℒ.
Since the universal canonical embedding identifies ωf with
𝒪ℙ(𝔼∨)(1)
restricted to the universal curve, restricting
along the section σi yields
With rational coefficients, ℓ
is tautological: indeed
$\ell=\frac{1}{5}c_1(\det\mathbb{E})=\frac{1}{5}\lambda_1$
on $\widetilde{\mathcal{Z}}$, and
$\mathrm{A}^*(\widetilde{\mathcal{Z}})\cong
\mathrm{A}^*(\mathcal{Z})$ because μ5 is
finite and we work with ℚ-coefficients.
Thus shows that the
hyperplane classes hi are
tautological on 𝒵.
The remaining classes originating on the base configuration stack
𝒱 are Chern
classes of the bundles determined by the stabilizer group G. In the simplest strata (the
independent locus 𝒰n) these are just Chern
classes of V; in the loci
with
canonical 8-subdivisors one also
encounters flag bundles such as W ⊂ U ⊂ V,
and
in the n = 12 overlap strata
one additionally has line bundles or rank-2 bundles encoding
hyperplane pencils or rulings on the associated scroll quadric. We now
explain how all such
classes are tautological.
The guiding point is that every linear subspace in ℙ4 = ℙ(H0(C, ωC)∨)
appearing in the stratum definition is determined by of canonical
sections at some of the marked points. Accordingly, the corresponding
stabilizer bundles are
identified with kernels (or images) of evaluation maps of the Hodge
bundle along marked points.
This reduces tautologicality of their Chern classes to
Grothendieck–Riemann–Roch computations
on the universal curve.
Let 𝒵 be as above, with universal
curve f: 𝒞𝒵 → 𝒵
and sections σi. For any
subset I ⊂ {1, …, n}
we set
𝒟I := ∑i ∈ Iσi(𝒵) ⊂ 𝒞𝒵, ω(−𝒟I) := ωf ⊗ 𝒪𝒞𝒵(−𝒟I).
Whenever we have imposed an open condition guaranteeing R1f*ω(−𝒟I) = 0
(which is the case in all configuration models we use, after removing
the relevant closed
degeneracy loci), the sheaf f*ω(−𝒟I)
is a vector bundle and fits into a
tautological exact sequence
n I}_i^*_f 0.
\end{equation}
In the geometric models, the kernels f*ω(−𝒟I)
are precisely the vector
bundles whose projectivizations parameterize linear subspaces determined
by the incidence data:
for instance, on a stratum where 4
points are required to lie on a plane Λ, the
corresponding rank-3 bundle W identifies with f*ω(−𝒟I0)
for the
appropriate 4-subset I0 of labels (those
constrained to lie on Λ);
similarly,
a hyperplane containing Λ
corresponds to a rank-4 kernel bundle
U for a
1-point or 4-point evaluation map, depending on the
chosen rigidification. In the
n = 12 binary overlap locus
𝒵I, J(2),
for example, the rigidified flag
W ⊂ U ⊂ V is
recovered from the marked-point data as successive kernels in
(first for the common 4 points I ∩ J, then after
specifying one
of the canonical hyperplanes). Likewise, in the ternary overlap locus
𝒵(3),
the ℙ1-parameter of ruling
planes is identified with a projectivization of
f*L,
where
$$
L\ :=\
\mathcal{O}_{\mathcal{C}_{\mathcal{Z}}}\Bigl(\sum_{i=1}^{12}\sigma_i\Bigr)\otimes
\omega_f^{-1},
$$
as in the discussion preceding Proposition~; here
R1f*L = 0
is ensured by the nontrigonal hypothesis together with the
base-point-free
condition in Lemma~.
Once a stabilizer bundle is expressed as f*ω(−𝒟I)
(or as a twist of such),
its Chern character is tautological by GRR:
ch(f*ω(−𝒟I)) = f*(ch(ω(−𝒟I)) ⋅ td(Tf)).
Expanding ch(ω(−𝒟I)) and
td(Tf) in
terms of
c1(ωf)
and the divisor classes σi, and then
pushing forward along f,
expresses each component of ch(f*ω(−𝒟I))
as a polynomial in
κ- and ψ-classes; equivalently, the
resulting Chern classes are tautological.
Since the Chern classes of any universal flag subbundle (such as W, U, and the
corresponding quotient bundles) can be obtained from these kernels by
repeated application of
, we conclude that Chern classes originating from the stabilizer
part of the configuration stack 𝒱 are
tautological on 𝒵.
We turn to the Grassmann part. Over 𝒱 we have, by construction, a quadratic
evaluation map
ev: Sym2V∨ → ⨁i = 1npi*𝒪ℙV(2),
and on the open locus where ev is
surjective its kernel E is a
vector bundle.
On G(3, E) we have the
universal exact sequence
where S is the universal
rank-3 subbundle. The Chow ring of
G(3, E) over
𝒱 is generated, as an A*(𝒱)-algebra, by the Chern
classes
ci(S)
(equivalently by those of Q);
restricting to the open substack 𝒳
does not introduce new generators.
It remains to see that the classes ci(S)
are tautological on 𝒵. By
it suffices to treat c(E), and c(E) is determined by
the
evaluation sequence
}^n p_i^*_{V}(2) 0.
\end{equation}
From we obtain
$$
c(E)\ =\ \frac{c(\mathrm{Sym}^2 V^\vee)}{\prod_{i=1}^n
c\bigl(p_i^*\mathcal{O}_{\mathbb{P}V}(2)\bigr)}
\ =\ \frac{c(\mathrm{Sym}^2 V^\vee)}{\prod_{i=1}^n (1+2h_i)}.
$$
Thus the Chern classes of E,
and hence those of S, are
polynomials in the Chern classes
of V and the hyperplane
classes hi. The classes
hi are
tautological by
. For V, we use 𝔼 ≅ V ⊗ ℒ on the
μ5-gerbe: since
c(𝔼) is the total λ-class and
c(ℒ) = 1 + ℓ with
ℓ = λ1/5,
the Chern classes of V are
polynomials
in λ1, …, λ5
(and in fact can be expressed by the standard formula
c(V) = c(𝔼 ⊗ ℒ−1)).
Consequently, c(E)
and c(S) are
tautological on 𝒵.
We record the conclusion in the form we use for localization.
We stress that Lemma~ does not depend on the
combinatorics of overlap types; it uses only the existence of a
presentation in which all
auxiliary parameters (planes, hyperplanes through a fixed plane, ruling
planes on a scroll, and
so on) are encoded by projective bundles attached to vector bundles that
arise from evaluation
kernels of the Hodge bundle or from pushforwards of line bundles
canonically constructed from
ωf and the
marked sections. This is precisely how the overlap models were
designed.
Once these inputs are in place, the Chow-theoretic identification of
generators proceeds
uniformly by , , GRR, and , exactly as
in the non-overlap strata.
As an immediate consequence of Lemma~, each stratum
𝒵 has tautological Chow, and the
pullback of A*(𝒵) to
any
of its Grassmann presentations is generated by λ- and ψ-classes. This is the
input needed for the localization and excision argument carried out in
the next section.
We now explain how the tautological generation established on each
locally closed stratum in
implies tautological generation on the whole nontrigonal locus
ℳ5, nnt := ℳ5, n \ ℳ5, n3,
and then on
ℳ5, n itself. The
argument is a formal application of the localization exact
sequence for Chow groups together with the fact that, on each stratum,
the Chow ring is generated
by restrictions of the global λ- and ψ-classes (Lemma~).
Let X be an algebraic stack
of finite type over k, and let
i: Z ↪ X
be a closed substack with open complement j: U := X \ Z ↪ X.
With
rational coefficients, Kresch’s Chow groups satisfy the standard
localization sequence (see
e.g. ):
We will use iteratively along a finite filtration by closed
substacks.
The key point is that we do not need to understand the map i* explicitly: it
suffices that
the Chow ring of U is
generated by restrictions of certain classes on X, so that any
class on U can be lifted to
X by choosing the same
polynomial in those generators.
We isolate the following formal lemma.
In practice, we apply Lemma~ with T the tautological ring of X,
generated by λ- and ψ-classes. The content of is
that,
for every stratum 𝒵 in our
stratification, A*(𝒵)
is
generated by classes that identify with polynomials in λ and ψ pulled back from
ℳ5, n; in
particular,
This is precisely the hypothesis needed to glue inductively along a
filtration by closed unions
of strata.
Fix n ≤ 12. On ℳ5, nnt, the
independent locus
𝒰n ⊂ ℳ5, nnt
is open by definition, and by
Theorem~B its complement is the union of the degeneracy loci DI indexed by
8-subsets
I ⊂ {1, …, n}:
ℳ5, nnt \ 𝒰n = ⋃|I| = 8DI.
For n ≤ 11, this already
yields a manageable decomposition: for n = 8, 9 the union is small,
and for n = 10, 11 the
components are pairwise disjoint on ℳ5, nnt
by
Lemma~. For n = 12, the
pairwise intersections DI ∩ DJ
are
nonempty along the tetragonal overlap locus (Lemma~), and we have
introduced a refinement of the stratification according to overlap
type
(Lemma~) together with Grassmann presentations of each overlap
stratum (Proposition~).
Choosing any linear extension of this partial order, we may arrange
the strata in a sequence
𝒵1, …, 𝒵N such
that, setting
Fr := ⋃s ≥ r𝒵s ⊂ ℳ5, nnt,
each Fr is
closed and Fr \ Fr + 1 = 𝒵r
(a disjoint union if several
strata have the same closure order, which we may incorporate by
reindexing). This produces a
finite decreasing closed filtration
Applying to each inclusion Fr + 1 ↪ Fr
gives exact
sequences
The advantage of is that it reduces statements about
A*(ℳ5, nnt)
to statements about A*(𝒵r)
for each stratum 𝒵r, together with the
compatibility of generators under
restriction.
We now deduce tautological generation on the full nontrigonal locus.
We now pass from the nontrigonal locus back to the full moduli stack.
Let
j: ℳ5, nnt ↪ ℳ5, n and i: ℳ5, n3 ↪ ℳ5, n
be the open immersion and its closed complement. Localization
gives
*} *(^{}_{5,n}) 0.
\end{equation}
By Proposition~, the rightmost term is tautological. Thus, to
deduce
tautologicality of A*(ℳ5, n), it
suffices to know that the Chow ring of
the trigonal locus is itself tautological, in a form compatible with
restriction of
λ- and ψ-classes.
We use the results of Canning–Larson on Chow and CKgP for
Brill–Noether type loci, specialized
to genus 5. Concretely, ℳ5, n3 is the
locus of curves admitting a
g31, and
contains the hyperelliptic locus ℳ5, n2 as a
closed substack.
The structure of these loci is controlled by Hurwitz stacks (degree
2 or 3
covers of
ℙ1 with marked points), from
which one obtains explicit presentations by quotient
stacks and projective bundles. In particular, Canning–Larson show that
the Chow rings of these
Hurwitz-type stacks are tautological and that their pushforwards to
ℳ5, n land in
the tautological ring; equivalently, the Chow rings of ℳ5, n2
and
ℳ5, n3
are generated by restrictions of λ- and ψ-classes from
ℳ5, n (with ℚ-coefficients).
We record this input in the form needed for gluing.
Assuming Proposition~, we can finish the tautological part of
Theorem~A by the same formal gluing as above.
We emphasize that, although the classes [DI] admit
explicit Thom–Porteous expressions
(Proposition~C), the localization argument above does not require them:
we never need to compute
the image of i* in
explicitly. What we use instead is the stronger
statement that A*(𝒵) is
generated by restrictions of global tautological
classes for each locally closed stratum 𝒵. This allows us to lift Chow classes
from a stratum to its ambient closed union simply by taking the same
polynomial in ψ and
λ, after which the difference
is supported on deeper strata and is treated
inductively.
With the tautological equality A*(ℳ5, n) = R*(ℳ5, n)
now
established for n ≤ 12, we
turn in the next section to the CKgP statement of Theorem~A,
which is proved by applying permanence properties of CKgP to the same
Grassmann/configuration
presentations and then gluing along the stratification.
We now prove the CKgP statement in Theorem~A. Throughout, we work
with rational Chow groups
in the sense of Kresch. Recall that a finite-type stack Y has CKgP if for every
quotient-stratified finite-type stack X the exterior product map
⊠: A*(Y)⊗ℚA*(X) → A*(Y × X)
is surjective. Our strategy parallels the tautological argument in
: we verify CKgP on each locally closed stratum via the explicit
Grassmann/configuration presentations, and then we glue along the finite
filtration by closed
unions of strata.
We collect the formal properties we use. They are established
in
Canning–Larson’s framework and amount to compatibilities of Chow groups
with
projective/Grassmann bundle formulas and with localization.
α in the image of A*(Y) ⊗ A*(X).
\end{proof}
Iterating Lemma~ along a finite decreasing closed filtration, we
obtain the
following convenient form.
We next verify CKgP for the basic building blocks that appear in the
Grassmann presentations of
our strata. The configuration stacks 𝒱
in Theorem~D are assembled from:
(i) classifying stacks BG of linear algebraic
groups G,
(ii) projective bundles ℙ(W) → BG for
G-representations W,
(iii) fiber products of such projective bundles over a common BG, and
(iv) open conditions removing diagonals and determinantal loci
corresponding to ``general
position’’ assumptions (e.g. distinct points, surjectivity of
evaluation, transversality).
All of these operations are compatible with quotient
stratifications.
We use the following input, which is standard in the CKgP formalism.
Fix n ≤ 12 and let 𝒵 ⊂ ℳ5, nnt be
any locally
closed stratum in the canonical-subset stratification (and, for n = 12, in the refined overlap
stratification). By Theorem~D (and by
Proposition~ in the overlap case), there exists a
configuration stack 𝒱, a vector bundle
E on 𝒱 defined as the kernel
of a surjective evaluation map on quadrics (after imposing an
appropriate open condition), and an
open immersion
𝒵 ↪ G(3, E)
whose image is cut out by transversality/smoothness conditions on the
corresponding net of
quadrics.
We now glue Proposition~ along the same finite filtration of
ℳ5, nnt
used in . Concretely, let
𝒵1, …, 𝒵N be
an ordering of strata as in
, and set Fr = ⋃s ≥ r𝒵s.
We finally pass from ℳ5, nnt to
ℳ5, n using
the
open–closed decomposition by the trigonal locus. Let
j: ℳ5, nnt ↪ ℳ5, n, i: ℳ5, n3 ↪ ℳ5, n
be the open immersion and closed complement.
Combining Theorem~ with Theorem~ completes the proof of
Theorem~A. We emphasize that, as for the tautological statement, the
CKgP argument is insensitive
to explicit cycle computations: once each locally closed stratum is
expressed as an open substack
of a Grassmann bundle over a configuration stack built from classifying
stacks and projective
bundles, CKgP follows formally from Lemmas~ and and
then propagates through the finite stratification.
In the case n = 12, the complement of the independent locus 𝒰12 inside the nontrigonal locus ℳ5, 12nt is the union of the degeneracy loci DI indexed by 8-subsets I ⊂ {1, …, 12}. In contrast with the cases n ≤ 11, these loci are not pairwise disjoint: there exist pointed curves (C; p1, …, p12) for which more than one 8-subset of marked points is canonical. The purpose of this appendix is to record the purely combinatorial constraints on such overlaps, to isolate a small list of overlap types, and to indicate where a computer enumeration is convenient for bookkeeping (even though the underlying patterns admit closed-form descriptions).
Fix a pointed curve (C; p) = (C; p1, …, p12) ∈ ℳ5, 12nt,
and set
Γ := p1 + ⋯ + p12.
For an 8-subset I ⊂ {1, …, 12}, let Ic denote its
complement, a 4-subset. We will
write
FI := ∑j ∈ Icpj,
so that Γ = ∑i ∈ Ipi + FI.
The condition (C; p) ∈ DI
is 𝒪C(∑i ∈ Ipi) ≅ ωC.
Equivalently,
𝒪C(FI) ≅ 𝒪C(Γ) ⊗ ωC−1.
Thus, for fixed (C; p), all
complements FI corresponding
to canonical 8-subsets are effective
divisors of degree 4 in a complete
linear system |𝒪C(Γ) ⊗ ωC−1|.
In particular, if I ≠ J and (C; p) ∈ DI ∩ DJ,
then FI ∼ FJ
as effective degree-4 divisors.
On ℳ5, 12nt we have the additional geometric input (Lemma~7.1 in the main text): the existence of two distinct canonical 8-subsets forces the resulting degree-4 divisors to move in a basepoint-free pencil g41. We record the immediate combinatorial consequence.
In terms of the 8-subsets
themselves, disjointness of complements is equivalent to a rigid
intersection size:
|I ∩ J| = 12 − |Ic ∪ Jc| = 12 − (4 + 4) = 4.
Thus, on the nontrigonal locus, any genuine overlap DI ∩ DJ
forces |I ∩ J| = 4,
and all other intersection sizes are excluded.
It is convenient to encode the possible overlaps by a graph. Let 𝒦 be the graph whose vertices are 4-subsets J ⊂ {1, …, 12}, with an edge between J and J′ if and only if J ∩ J′ = ⌀. This is the Kneser graph KG(12, 4).
Given (C; p),
define
𝒮(C; p) := { I ⊂ {1, …, 12} : |I| = 8, (C; p) ∈ DI }.
Equivalently, 𝒮(C; p) corresponds
to the set of complements
𝒯(C; p) := { Ic ⊂ {1, …, 12} : I ∈ 𝒮(C; p) },
a collection of 4-subsets.
Lemma~ says that 𝒯(C; p) is a clique in 𝒦, i.e. a family of pairwise disjoint 4-subsets. Since {1, …, 12} has size 12, such a family has cardinality at most 3. We obtain:
In particular, on the nontrigonal locus there are only three overlap
cardinalities to consider:
|𝒮(C; p)| ∈ {0, 1, 2, 3}.
The case |𝒮| = 0 is precisely 𝒰12. The case |𝒮| = 1 corresponds to the open part of a
single DI
away from all other DJ. The cases
|𝒮| = 2 and |𝒮| = 3 are the genuine overlap strata.
It is also useful to note that the closure relations among the strata
are governed by inclusion of cliques: the locus |𝒮| ≥ 3 is contained in the locus |𝒮| ≥ 2, which is contained in ⋃IDI.
Concretely, for a partition {1, …, 12} = J1 ⊔ J2 ⊔ J3
into 4-subsets, the associated three
8-subsets Ia := Jb ⊔ Jc
satisfy
DI1 ∩ DI2 ∩ DI3 ⊂ DIa ∩ DIb ⊂ DIa.
Because the markings are ordered, the stratification used in the main argument is indexed by the actual subsets I ⊂ {1, …, 12}. Nevertheless, up to the natural action of 𝔖12 permuting labels, there are very few combinatorial types. We list them in a form adapted to constructing the refined filtration for localization.
No canonical 8-subset occurs:
𝒮(C; p) = ⌀.
This is (C; p) ∈ 𝒰12.
Exactly one 8-subset I is canonical:
𝒮(C; p) = {I}.
Equivalently, a single 4-subset J = Ic
occurs as FI, and no other
4-subset of marked points lies in the
same degree-4 linear system |𝒪C(Γ) ⊗ ωC−1|
as an effective divisor.
Exactly two canonical 8-subsets occur.
By Proposition~ this is the same as choosing two disjoint 4-subsets J1, J2 ⊂ {1, …, 12}
and setting
𝒮(C; p) = { I1, I2 }, Ia := {1, …, 12} \ Ja.
Writing J3 := {1, …, 12} \ (J1 ⊔ J2),
we have
I1 = J2 ⊔ J3, I2 = J1 ⊔ J3, |I1 ∩ I2| = |J3| = 4.
Geometrically (again by Lemma~7.1), the divisors ∑j ∈ J1pj
and ∑j ∈ J2pj
are two fibers of a g41 on C, while the remaining 4 marked points indexed by J3 are unconstrained by
the overlap condition beyond the exclusions built into ℳ5, 12nt.
Exactly three canonical 8-subsets
occur. Then {1, …, 12} is partitioned
into three 4-subsets J1, J2, J3,
and
𝒮(C; p) = { I1, I2, I3 }, Ia := {1, …, 12} \ Ja.
Equivalently,
I1 = J2 ⊔ J3, I2 = J1 ⊔ J3, I3 = J1 ⊔ J2.
In this case the three degree-4
divisors ∑j ∈ Japj
are three fibers of a g41. This is
the extremal overlap allowed on ℳ5, 12nt.
For later use, we record the number of labeled instances of Types
1–3, i.e. the number of
distinct indexing choices in {1, …, 12}:
$$
\#\{\,I:\ |I|=8\,\}\ =\ \binom{12}{8}\ =\ 495,
$$
$$
\#\{\,\{I,J\}:\ |I|=|J|=8,\ I\neq J,\ I^c\cap J^c=\varnothing\,\}
\ =\ \frac{1}{2}\binom{12}{4}\binom{8}{4}\ =\ 17325,
$$
$$
\#\{\,\{I_1,I_2,I_3\}:\ |I_a|=8,\ I_a^c\ \text{partition}\
\{1,\dots,12\}\,\}
\ =\ \frac{1}{6}\binom{12}{4}\binom{8}{4}\ =\ 5775.
$$
These counts are useful when one implements the refined overlap
stratification as a finite filtration by closed unions: one must know
that the relevant index sets are finite and have no hidden additional
patterns.
We briefly discuss the expected codimension of each type inside ℳ5, 12nt, both as a sanity check and as a guide to identifying the correct ``smooth open’’ inside the Grassmann models.
First, dim ℳ5, 12 = (3 ⋅ 5 − 3) + 12 = 24,
hence dim ℳ5, 12nt = 24 as
well. For any fixed 8-subset I, the degeneracy locus DI is the
rank- ≤ 4 locus of a morphism of
bundles of ranks 5 → 8, and the
Thom–Porteous expected codimension is
(5 − 4)(8 − 4) = 4.
Thus dim DI is
expected to be 20, and on the
nontrigonal locus this is indeed the relevant value.
For a pair overlap DI ∩ DJ
with I ≠ J,
Proposition~ forces Ic ∩ Jc = ⌀,
hence I ∩ J has size
4. From the point of view of parameters
of pointed curves, the conditions
\(\sum_{i\in I}p_i\) is canonical'' and∑j ∈ Jpj
is canonical’’ are imposed on two different 8-tuples sharing only the 4 points indexed by I ∩ J. One therefore
expects these two codimension-4
conditions to meet properly on ℳ5, 12nt, giving
expected codimension 8, i.e. expected
dimension 16. Any potential excess
would arise from the possibility that the two canonicality conditions
are not independent; on the nontrigonal locus this is precisely what
Lemma~7.1 rules out by forcing a g41 rather
than a basepointed series (which would drop the effective codimension
but would also force trigonal behavior).
Similarly, in the triple overlap case the expected codimension is 12, giving expected dimension 12. This matches the heuristic geometric picture: for a given curve C (dimension 12), requiring that 12 marked points form three fibers of a fixed g41 is essentially a discrete condition once the three target points of ℙ1 are chosen (a 3-dimensional choice), but the ordering of points within each fiber introduces only finite étale ambiguity. The resulting dimension 12 + 3 on the universal pointed curve is consistent with codimension 12 in dimension 24.
In practice, our refined stratification uses pieces, e.g.
DI∘ := DI \ ⋃J ≠ IDJ, (DI ∩ DJ)∘ := (DI ∩ DJ) \ ⋃K ∉ {I, J}DK,
and similarly for triple intersections. The expected codimension
discussion above should be interpreted for the closures DI, DI ∩ DJ,
and DI1 ∩ DI2 ∩ DI3;
the open parts are obtained by excising further closed unions of
lower-dimensional overlap loci.
Although the overlap patterns are rigid, a computer check can be convenient in two places.
When implementing a localization argument with a filtration by closed
unions of strata, one needs a compatible ordering and a verification
that each successive difference is a disjoint union of prescribed
locally closed pieces. Concretely, one wants to know, for each fixed
I, which J can occur with DI ∩ DJ ≠ ⌀,
and for each pair (I, J), which K can occur with DI ∩ DJ ∩ DK ≠ ⌀.
On ℳ5, 12nt the
answer is ``exactly those with disjoint complements’’, but it is still
useful to have an explicit adjacency list for the 495 indices I, and an explicit list of the 5775 triples, to avoid omissions.
If one ignores the nontrigonal hypothesis, there are many set-theoretic
possibilities for intersections of 8-subsets (e.g. |I ∩ J| ≥ 5), and it is
easy to inadvertently allow an overlap type which in fact forces
trigonal or hyperelliptic behavior. A quick enumeration of all pairs (or
triples) of 8-subsets and their
intersection sizes provides a mechanical check that, after imposing
Lemma~7.1, only the patterns listed above remain. Equivalently, one
checks that all cliques in KG(12, 4)
have size at most 3, and that
size-3 cliques are precisely partitions
into three 4-subsets.
We stress that the geometric input is indispensable: purely combinatorially, $\binom{12}{8}$ subsets admit many overlap patterns, but the basepoint-freeness implicit in Lemma~7.1 collapses the possibilities to the three types above. This is exactly what makes the refined overlap stratification finite and manageable, and it is the combinatorial shadow of the tetragonal/scroll geometry used to produce explicit configuration-stack models for the overlap strata.
We record here a concrete instance of the general determinantal
computations used in the main argument. Throughout we work on the
nontrigonal locus
ℳ5, 12nt := ℳ5, 12 \ ℳ5, 123,
so that the only source of failure of independence for quadrics is the
presence of a canonical effective subdivisor supported on the marked
points.
Fix an 8-subset I ⊂ {1, …, 12}. Let f : 𝒞5, 12 → ℳ5, 12
be the universal curve and σi the sections.
Write
𝔼 := f*ωf
for the Hodge bundle (rank 5 in genus
5), and for each i set
Li := σi*ωf, c1(Li) = ψi.
Define the rank-8 bundle
𝔽I := ⨁i ∈ ILi, c(𝔽I) = ∏i ∈ I(1 + ψi).
The locus DI ⊂ ℳ5, 12nt
is the degeneracy locus where the natural evaluation morphism
φI : 𝔼 → 𝔽I
has rank ≤ 4; equivalently, where
∑i ∈ Ipi
is a canonical divisor. Since rk(𝔼) = 5
and rk(𝔽I) = 8, the
expected codimension of the rank- ≤ 4
locus is
(5 − 4)(8 − 4) = 4,
and on ℳ5, 12nt
this is the relevant codimension. Thus the Thom–Porteous formula
gives
To make~ completely explicit, we expand c(𝔽I − 𝔼) = c(𝔽I) c(𝔼)−1
up to degree 4. Write
c(𝔼) = 1 + λ1 + λ2 + λ3 + λ4 + λ5,
and let qj
denote the degree-j component
of c(𝔼)−1. A
standard formal inversion yields
Next, let ea(ψI)
be the elementary symmetric polynomial of degree a in the variables {ψi}i ∈ I,
so that
$$
\prod_{i\in I}(1+\psi_i)=\sum_{a=0}^8 e_a(\psi_I),
\qquad e_0(\psi_I)=1,\quad e_1(\psi_I)=\sum_{i\in I}\psi_i,\ \ldots
$$
Then the degree-4 component of c(𝔽I) c(𝔼)−1
is
$$
c_4(\mathbb{F}_I-\mathbb{E})
=\sum_{j=0}^4 e_{4-j}(\psi_I)\,q_j,
$$
and we obtain the explicit tautological polynomial
In particular, [DI] lies in the
subring generated by λ- and
ψ-classes, hence is
tautological. We emphasize that~ holds for every choice of I (with the evident dependence on
the variables {ψi}i ∈ I).
For a concrete instance, take I = {1, …, 8}. Then ea(ψI) is the elementary symmetric polynomial in ψ1, …, ψ8, and [D{1, …, 8}] is obtained by substituting those ea into~. In codimension 4 one may also expand ea in terms of monomials if desired (e.g. e2 = ∑1 ≤ i < j ≤ 8ψiψj, etc.), but for bookkeeping in localization arguments it is typically preferable to retain the symmetric notation.
We next treat a representative overlap situation of Type 2 from Appendix~. Fix a
partition of {1, …, 12} into three
4-subsets
J1 = {1, 2, 3, 4}, J2 = {5, 6, 7, 8}, J3 = {9, 10, 11, 12},
and set the associated 8-subsets
I1 := J2 ⊔ J3 = {5, 6, 7, 8, 9, 10, 11, 12}, I2 := J1 ⊔ J3 = {1, 2, 3, 4, 9, 10, 11, 12},
so that I1c = J1
and I2c = J2
are disjoint. Consider the closed intersection
Z12(J1, J2) := DI1 ∩ DI2 ⊂ ℳ5, 12nt.
By the combinatorial constraints recorded earlier (ultimately stemming
from Lemma~7.1 in the main text), this intersection is precisely the
locus where two distinct canonical 8-subsets occur with disjoint complements:
equivalently, where the effective degree-4 divisors
FI1 = p1 + p2 + p3 + p4, FI2 = p5 + p6 + p7 + p8
lie in the same basepoint-free g41 on C. The corresponding overlap stratum
of Type 2 is obtained
by removing the (possibly empty) triple-overlap locus where also I3 := J1 ⊔ J2
is canonical:
Z12∘(J1, J2) := (DI1 ∩ DI2) \ DI3.
Since DI3 ∩ DI1 ∩ DI2
has codimension 12 (when nonempty)
whereas DI1 ∩ DI2
has codimension 8, the removal does not
change the codimension-8 cycle class in
the ambient Chow ring:
[Z12∘(J1, J2)] = [Z12(J1, J2)] ∈ A8(ℳ5, 12nt).
Thus it suffices to compute the class of the closed intersection.
On ℳ5, 12nt,
the loci DI1
and DI2
have the expected codimension 4, and
their intersection has the expected codimension 8 (cf. the consistency check in Appendix~).
In this situation the intersection product in Chow computes the
fundamental class:
Combining~ with~ gives an explicit tautological expression. For
compactness, set
Ea(1) := ea(ψI1)
in
{ψ5, ψ6, ψ7, ψ8, ψ9, ψ10, ψ11, ψ12}, Ea(2) := ea(ψI2)
in
{ψ1, ψ2, ψ3, ψ4, ψ9, ψ10, ψ11, ψ12}.
Also retain q0, …, q4
as above. Then
$$
[D_{I_\nu}]=\sum_{j=0}^4 E_{4-j}^{(\nu)}\,q_j
\qquad (\nu=1,2),
$$
and hence
Expanding qaqb
if desired, one obtains a polynomial in λ1, …, λ4
with coefficients in the subring generated by the ψi. For
instance, the λ-free term
(degree 0 in the λ’s) is simply
E4(1)E4(2),
and the coefficient of λ1 is
−(E3(1)E4(2) + E4(1)E3(2))λ1,
while higher λ-degrees mix in
the expected way dictated by q2, q3, q4.
The main point for our applications is that~ is already an explicit,
finite tautological formula, uniform in the indexing data.
For completeness we also record the analogous expression for the Type
3 (triple overlap)
closure associated to the above partition. With I3 = J1 ⊔ J2 = {1, …, 8},
one has
[DI1 ∩ DI2 ∩ DI3] = [DI1] ⋅ [DI2] ⋅ [DI3] ∈ A12(ℳ5, 12nt),
and [DI3]
is again given by~ with ψ1, …, ψ8
as variables. In particular, both the pair-overlap and triple-overlap
closures admit tautological cycle classes written down explicitly in
terms of λ- and ψ-classes, which is the input needed
to implement the localization filtration on ℳ5, 12nt by closed
unions of overlap strata.