arXiv subjects

Samir Canning

Publications and source records attributed to Samir Canning.

At least 19 recordsLinked to original sources

arXiv2026

The Gorenstein property and Pixton's conjecture for compact type moduli

We show that the tautological ring of $\mathcal{M}_{g,n}^{\mathrm{ct}}$ is not Gorenstein for $g\geq 2$ and $2g+n\geq 12$. We prove new cases of Pixton's conjecture that the $3$-spin relations are a complete set of relations for the tautological ring, including $\mathcal{M}_{6}^{\mathrm{ct}}$, $\mathcal{M}_{5,2}^{\mathrm{ct}}$, and $\mathcal{M}_7^{\mathrm{ct}}$. These are the first known cases where Pixton's conjecture is true, but the tautological ring is not Gorenstein. These results are also a key ingredient in recent work on non-tautological cycles on the moduli space of principally polarized abelian varieties.

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arXiv2026

The low degree cohomology of compactifications of $A_g$

We compute the low degree $\ell$-adic intersection cohomology of symplectic local systems on the Satake compactification of the moduli space $A_g$ of principally polarized abelian varieties. We prove that only a small finite list of irreducible Galois representations can appear in the low degree cohomology of any nonsingular toroidal compactification of $A_g$ or $X_{g,s}$, the $s$-fold fiber product of the universal abelian variety. We give several applications, including to spaces of holomorphic forms on toroidal compactifications and to the cohomology of the interior. In particular, we give a complete characterization of when the cohomology of $X_{g,s}$, or one of its toroidal compactifications, is of Tate type. The result is independent of the choice of toroidal compactification.

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arXiv2026

Torelli loci, product cycles, and the homomorphism conjecture for $\mathcal{A}_g$

The tautological $\mathbb{Q}$-subalgebra $\mathsf{R}^*(\mathcal{A}_g) \subset \mathsf{CH}^*(\mathcal{A}_g)$ of the Chow ring of the moduli space of principally polarized abelian varieties is generated by the Chern classes of the Hodge bundle. There is a canonical $\mathbb{Q}$-linear projection operator $\mathsf{taut}: \mathsf{CH}^*(\mathcal{A}_g) \rightarrow \mathsf{R}^*(\mathcal{A}_g).$ We present here new calculations of intersection products of the Torelli locus in $\mathcal{A}_g$ with the product loci $\mathcal{A}_{r}\times \mathcal{A}_{g-r} \rightarrow \mathcal{A}_g$ for $r\leq 3$. The results suggest that $\mathsf{taut}$ is a $\mathbb{Q}$-algebra homomorphism, at least for special cycles. We discuss a conjectural framework for this homomorphism property. Our calculations follow two independent approaches. The first is a direct study of the excess intersection geometry of the fiber product of the Torelli and product morphisms. The second recasts the geometry in terms of families Gromov-Witten classes, which are computed by a wall-crossing formula related to unramified maps. We define tautological projections of cycles on the fiber products $\mathcal X_g^s \to \mathcal A_g$ of the universal family. We compute these projections for a class of product cycles on $\mathcal X_g^s$ in terms of a determinant involving the universal theta divisors and Poincar\'e classes. Using Abel-Jacobi pullbacks of product cycles on $\mathcal X_g^s$ and their projections, we construct a new family of classes which we conjecture to lie in the Gorenstein kernels of the tautological rings $\mathsf{R}^*(\mathcal M^{\mathrm{ct}}_{g,n})$. In particular, we construct nontrivial elements of the Gorenstein kernels of $\mathsf{R}^5(\mathcal{M}_{5,2}^{\mathrm{ct}})$ and $\mathsf{R}^5(\mathcal{M}_{4,4}^{\mathrm{ct}})$.

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arXiv2025

FA-modules of holomorphic forms on $\overline{\mathcal{M}}_{g,n}$

For fixed genus g and varying finite marking set A, the gluing and forgetful maps give the spaces of holomorphic forms on the moduli space of stable A-marked curves of genus g has the structure of an FA-module, i.e., a functor from the category of finite sets to vector spaces. We prove that the resulting FA-modules of holomorphic k-forms are simple, for k less than or equal to 18, whenever they are nonzero. Conditional upon the conjectured vanishing of holomorphic 19-forms and 20-forms in genus 3, for 15 and 16 marked points, respectively, this extends to k less than or equal to 20.

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arXiv2024

The motivic structures $\mathsf{LS}_{12}$ and $\mathsf{S}_{16}$ in the cohomology of moduli spaces of curves

We study the appearances of $\mathsf{LS}_{12}$ and $\mathsf{S}_{16}$ in the weight-graded compactly supported cohomology of moduli spaces of curves. As applications, we prove new nonvanishing results for the middle cohomology groups of $\mathcal{M}_9$ and $\mathcal{M}_{11}$ and give evidence to support the conjecture that the dimension fo $H^{2g + k}_c(\mathcal{M}_g)$ grows at least exponentially with $g$ for almost all $k$.

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arXiv2024

Moduli spaces of curves with polynomial point counts

We prove that the number of curves of a fixed genus g over finite fields is a polynomial function of the size of the field if and only if g is at most 8. Furthermore, we determine for each positive genus g the smallest n such that the moduli space of curves of genus g with n marked points does not have polynomial point count. A key ingredient in the proofs, which is also a new result of independent interest, is the computation of the thirteenth cohomology group of the moduli spaces of stable curves of genus g with n marked points, for all g and n.

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arXiv2024

Tautological and non-tautological cycles on the moduli space of abelian varieties

The tautological Chow ring of the moduli space $\mathcal{A}_g$ of principally polarized abelian varieties of dimension $g$ was defined and calculated by van der Geer in 1999. By studying the Torelli pullback of algebraic cycles classes from $\mathcal{A}_g$ to the moduli space $\mathcal{M}_g^{\mathrm{ct}}$ of genus $g$ of curves of compact type, we prove that the product class $[\mathcal{A}_1\times \mathcal{A}_5]\in \mathsf{CH}^{5}(\mathcal{A}_6)$ is non-tautological, the first construction of an interesting non-tautological algebraic class on the moduli spaces of abelian varieties. For our proof, we use the complete description of the the tautological ring $\mathsf{R}^*(\mathcal{M}_6^{\mathrm{ct}})$ in genus 6 conjectured by Pixton and recently proven by Canning-Larson-Schmitt. The tautological ring $\mathsf{R}^*(\mathcal{M}_6^{\mathrm{ct}})$ has a 1-dimensional Gorenstein kernel, which is geometrically explained by the Torelli pullback of $[\mathcal{A}_1\times \mathcal{A}_5]$. More generally, the Torelli pullback of the difference between $[\mathcal{A}_1\times \mathcal{A}_{g-1}]$ and its tautological projection always lies in the Gorenstein kernel of $\mathsf{R}^*(\mathcal{M}_g^{\mathrm{ct}})$. The product map $\mathcal{A}_1\times \mathcal{A}_{g-1}\rightarrow \mathcal{A}_g$ is a Noether-Lefschetz locus with general Neron-Severi rank 2. A natural extension of van der Geer's tautological ring is obtained by including more general Noether-Lefschetz loci. Results and conjectures related to cycle classes of Noether-Lefschetz loci for all $g$ are presented.

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arXiv2024

The tautological ring of $\overline{\mathcal{M}}_{g,n}$ is rarely Gorenstein

We prove that the tautological rings $\mathsf{R}^*(\overline{\mathcal{M}}_{g,n})$ and $\mathsf{RH}^*(\overline{\mathcal{M}}_{g,n})$ are not Gorenstein when $g\geq 2$ and $2g+n\geq 24$, extending results of Petersen and Tommasi in genus $2$. The proof uses the intersection of tautological classes with non-tautological bielliptic cycles. We conjecture the converse: the tautological rings should be Gorenstein when $g=0,1$ or $g\geq 2$ and $2g+n<24$. The conjecture is known for $g=0,1$ by work of Keel and Petersen, and we prove several new cases of this conjecture for $\mathsf{RH}^*(\overline{\mathcal{M}}_{g,n})$ when $g\geq 2$.

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arXiv2024

Holomorphic forms and non-tautological cycles on moduli spaces of curves

We prove, for infinitely many values of $g$ and $n$, the existence of non-tautological algebraic cohomology classes on the moduli space $\mathcal{M}_{g,n}$ of smooth, genus-$g$, $n$-pointed curves. In particular, when $n=0$, our results show that there exist non-tautological algebraic cohomology classes on $\mathcal{M}_g$ for $g=12$ and all $g \geq 16$. These results generalize the work of Graber--Pandharipande and van Zelm, who proved that the classes of particular loci of bielliptic curves are non-tautological and thereby exhibited the only previously-known non-tautological class on any $\mathcal{M}_g$: the bielliptic cycle on $\mathcal{M}_{12}$. We extend their work by using the existence of holomorphic forms on certain moduli spaces $\overline{\mathcal{M}}_{g,n}$ to produce non-tautological classes with nontrivial restriction to the interior, via which we conclude that the classes of many new double-cover loci are non-tautological.

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arXiv2024

Tautological projection for cycles on the moduli space of abelian varieties

We define a tautological projection operator for algebraic cycle classes on the moduli space of principally polarized abelian varieties $\mathcal{A}_g$: every cycle class decomposes canonically as a sum of a tautological and a non-tautological part. The main new result required for the definition of the projection operator is the vanishing of the top Chern class of the Hodge bundle over the boundary $\bar{\mathcal{A}}_g\smallsetminus \mathcal{A}_g$ of any toroidal compactification $\bar{\mathcal{A}}_g$ of the moduli space $\mathcal{A}_g$. We prove the vanishing by a careful study of residues in the boundary geometry. The existence of the projection operator raises many natural questions about cycles on $\mathcal{A}_g$. We calculate the projections of all product cycles $\mathcal{A}_{g_1}\times \ldots \times \mathcal{A}_{g_\ell}$ in terms of Schur determinants, discuss Faber's earlier calculations related to the Torelli locus, and state several open questions. The Appendix contains a conjecture about the projection of the locus of abelian varieties with real multiplication.

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arXiv2023

The Chow ring of the moduli space of degree $2$ quasi-polarized K3 surfaces

We study the Chow ring with rational coefficients of the moduli space $\mathcal F_{2}$ of quasi-polarized $K3$ surfaces of degree $2$. We find generators, relations, and calculate the Chow Betti numbers. The highest nonvanishing Chow group is $\mathsf A^{17}(\mathcal F_2)\cong {\mathbb{Q}}$. We prove that the Chow ring consists of tautological classes and is isomorphic to the even cohomology. The Chow ring is not generated by divisors and does not satisfy duality with respect to the pairing into $\mathsf A^{17}(\mathcal F_2)$. The kernel of the pairing is a 1-dimensional subspace of $\mathsf{A}^{9}(\mathcal F_2)$ which we calculate explicitly. In the appendix, we revisit Kirwan-Lee's calculation of the Poincar\'e polynomial of $\mathcal{F}_2$.

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arXiv2023

Extensions of tautological rings and motivic structures in the cohomology of $\overline{\mathcal{M}}_{g,n}$

We study collections of subrings of $H^*(\overline{\mathcal{M}}_{g,n})$ that are closed under the tautological operations that map cohomology classes on moduli spaces of smaller dimension to those on moduli spaces of larger dimension and contain the tautological subrings. Such extensions of tautological rings are well-suited for inductive arguments and flexible enough for a wide range of applications. In particular, we confirm predictions of Chenevier and Lannes for the $\ell$-adic Galois representations and Hodge structures that appear in $H^k(\overline{\mathcal{M}}_{g,n})$ for $k = 13$, $14$, and $15$. We also show that $H^4(\overline{\mathcal{M}}_{g,n})$ is generated by tautological classes for all $g$ and $n$, confirming a prediction of Arbarello and Cornalba from the 1990s. In order to establish the final bases cases needed for the inductive proofs of our main results, we use Mukai's construction of canonically embedded pentagonal curves of genus 7 as linear sections of an orthogonal Grassmannian and a decomposition of the diagonal to show that the pure weight cohomology of $\mathcal{M}_{7,n}$ is generated by algebraic cycle classes, for $n \leq 3$.

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arXiv2022

The bielliptic locus in genus 11

The Chow ring of $\mathcal{M}_g$ is known to be generated by tautological classes for $g \leq 9$. Meanwhile, the first example of a non-tautological class on $\mathcal{M}_{g}$ is the fundamental class of the bielliptic locus in $\mathcal{M}_{12}$, due to van Zelm. It remains open if the Chow rings of $\mathcal{M}_{10}$ and $\mathcal{M}_{11}$ are generated by tautological classes. In these cases, a natural first place to look is at the bielliptic locus. In genus $10$, it is already known that classes supported on the bielliptic locus are tautological. Here, we prove that all classes supported on the bielliptic locus are tautological in genus $11$. By Looijenga's vanishing theorem, this implies that they all vanish.

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arXiv2022

The eleventh cohomology group of $\bar{\mathcal{M}}_{g,n}$

We prove that the rational cohomology group $H^{11}(\bar{\mathcal{M}}_{g,n})$ vanishes unless $g = 1$ and $n \geq 11$. We show furthermore that $H^k(\bar{\mathcal{M}}_{g,n})$ is pure Hodge-Tate for all even $k \leq 12$ and deduce that $\# \bar{\mathcal{M}}_{g,n}(\mathbb{F}_q)$ is surprisingly well approximated by a polynomial in $q$. In addition, we use $H^{11}(\bar{\mathcal{M}}_{1,11})$ and its image under Gysin push-forward for tautological maps to produce many new examples of moduli spaces of stable curves with nonvanishing odd cohomology and non-tautological algebraic cycle classes in Chow cohomology.

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arXiv2022

On the Chow and cohomology rings of moduli spaces of stable curves

In this paper, we ask: for which $(g, n)$ is the rational Chow or cohomology ring of $\overline{\mathcal{M}}_{g,n}$ generated by tautological classes? This question has been fully answered in genus $0$ by Keel (the Chow and cohomology rings are tautological for all $n$) and genus $1$ by Belorousski (the rings are tautological if and only if $n \leq 10$). For $g \geq 2$, work of van Zelm shows the Chow and cohomology rings are not tautological once $2g + n \geq 24$, leaving finitely many open cases. Here, we prove that the Chow and cohomology rings of $\overline{\mathcal{M}}_{g,n}$ are isomorphic and generated by tautological classes for $g = 2$ and $n \leq 9$ and for $3 \leq g \leq 7$ and $2g + n \leq 14$. For such $(g, n)$, this implies that the tautological ring is Gorenstein and $\overline{\mathcal{M}}_{g,n}$ has polynomial point count.

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arXiv2022

The rational Chow rings of moduli spaces of hyperelliptic curves with marked points

We determine the rational Chow ring of the moduli space $\mathcal{H}_{g,n}$ of $n$-pointed smooth hyperelliptic curves of genus $g$ when $n \leq 2g+6$. We also show that the Chow ring of the partial compactification $\mathcal{I}_{g,n}$, parametrizing $n$-pointed irreducible nodal hyperelliptic curves, is generated by tautological divisors. Along the way, we improve Casnati's result that $\mathcal{H}_{g,n}$ is rational for $n \leq 2g+8$ to show $\mathcal{H}_{g,n}$ is rational for $n \leq 3g+5$.

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arXiv2022

The integral Chow rings of moduli of Weierstrass fibrations

We compute the Chow rings with integral coefficients of moduli stacks of minimal Weierstrass fibrations over the projective line. For each integer $N\geq 1$, there is a moduli stack $\mathcal{W}^{\mathrm{min}}_N$ parametrizing minimal Weierstrass fibrations with fundamental invariant $N$. Following work of Miranda and Park--Schmitt, we give a quotient stack presentation for each $\mathcal{W}^{\mathrm{min}}_N$. Using these presentations and equivariant intersection theory, we determine a complete set of generators and relations for each of the Chow rings. For the cases $N=1$ (respectively, $N=2$), parametrizing rational (respectively, K3) elliptic surfaces, we give a more explicit computation of the relations.

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arXiv2021

Hyperk\"ahler manifolds

We give an elementary introduction to hyperk\"ahler manifolds, survey some of their interesting properties and some open problems.

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