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Hannah Larson

Publications and source records attributed to Hannah Larson.

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

Chow rings, cohomology rings, and point counts of moduli spaces of curves

In this expository article, we present on state-of-the art results regarding three closely related invariants of moduli spaces of curves: their Chow rings, cohomology rings, and point counts over finite fields. We study the moduli space $\mathcal{M}_{g,n}$, parameterizing smooth genus $g$ curves with $n$ marked points, as well as its compactification by stable curves $\overline{\mathcal{M}}_{g,n}$. After explaining the relationship between these different invariants, we survey what is know regarding the following related questions: When are the Chow rings tautological? When are the cohomology groups tautological? And when are the point counts over fields of size $q$ given by a polynomial in $q$?

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arXiv2026

Independence of tautological classes and cohomological stability for strata of differentials

The tautological rings of strata of differentials are known to be generated by divisor classes. In this paper, we give lower bounds on the degrees of relations among them, depending on the genus $g$ and the number of simple zeros. For strata with more than $4g/3$ simple zeros, our results show that there are no relations in degrees less than $\lfloor g/3 \rfloor + 1$. Moreover, we conjecture that, outside of a few exceptions, there is always a non-trivial relation in degree $\lfloor g/3 \rfloor + 1$, and prove the conjecture for all strata of holomorphic abelian differentials with $g \leq 30$. We also prove that the cohomology rings of strata of holomorphic differentials with sufficiently many simple zeros stabilize to the free algebra on the tautological divisor class. Finally, we show that for a large class of holomorphic abelian strata, containing hyperelliptic differentials, the tautological ring is non-trivial for sufficiently large $g$.

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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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arXiv2025

Chow Rings of Hurwitz Spaces with Marked Ramification

The Hurwitz space $\overline{\mathscr{H}}_{k,g}$ is a compactification of the space of smooth genus-$g$ curves with a simply-branched degree-$k$ map to $\mathbb{P}^1$. In this paper, we initiate a study of the Chow rings of these spaces, proving in particular that when $k=3$ (which is the first case in which the Chow ring is not already known), the codimension-2 Chow group is generated by the fundamental classes of codimension-2 boundary strata. The key tool is to realize the codimension-1 boundary strata of $\overline{\mathscr{H}}_{3,g}$ as the images of gluing maps whose domains are products of Hurwitz spaces $\mathscr{H}_{k',g'}(\mu)$ with a single marked fiber of prescribed (not necessarily simple) ramification profile $\mu$, and to prove that the spaces $\mathscr{H}_{k',g'}(\mu)$ with $k'=2,3$ have trivial Chow ring.

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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

Complex Bott Periodicity in algebraic geometry

We state and prove a form of Bott periodicity (for $U(n)$) in an algebraic setting (so, $GL(n)$) which makes sense over $\mathbb{Z}$, which also specializes to Bott periodicity in the usual sense (hence giving yet another proof of classical Bott periodicity). An appendix by B. Church gives a specialization of the constructions and results to motivic homotopy theory, which may be of independent interest.

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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

Brill--Noether theory of smooth curves in the plane and on Hirzebruch surfaces

In this paper, we describe the Brill--Noether theory of a general smooth plane curve and a general curve $C$ on a Hirzebruch surface of fixed class. It is natural to study the line bundles on such curves according to the splitting type of their pushforward along projection maps $C \to \mathbb{P}^1$. Inspired by Wood's parameterization of ideal classes in rings associated to binary forms, we further refine the stratification of line bundles $L$ on $C$ by fixing the splitting types of both $L$ and $L(\Delta)$, where $\Delta \subset C$ is the intersection of $C$ with the directrix of the Hirzebruch surface. Our main theorem determines the dimensions of these locally closed strata and, if the characteristic of the ground field is zero, proves that they are smooth.

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arXiv2024

The Chow ring of the universal Picard stack over the hyperelliptic locus

Let $\mathscr{J}^d_g \to \mathscr{M}_g$ be the universal Picard stack parametrizing degree $d$ line bundles on genus $g$ curves, and let $\mathscr{J}^d_{2,g}$ be its restriction to locus of hyperelliptic curves $\mathscr{H}_{2,g} \subset \mathscr{M}_g$. We determine the rational Chow ring of $\mathscr{J}^d_{2,g}$ for all $d$ and $g$. In particular, we prove it is generated by restrictions of tautological classes on $\mathscr{J}^d_g$ and we determine all relations among the restrictions of such classes. We also compute the integral Picard group of $\mathscr{J}^d_{2,g}$, completing (and extending to the $\mathrm{PGL}_2$-equivariant case) prior work of Erman and Wood. As a corollary, we prove that $\mathscr{J}^d_{2,g}$ is either a trivial $\mathbb{G}_m$-gerbe over its rigidification, or has Brauer class of order $2$, depending on the parity of $d - g$.

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arXiv2023

Maximal Brill--Noether loci via the gonality stratification

We study the restriction of Brill-Noether loci to the gonality stratification of the moduli space of curves of fixed genus. As an application, we give new proofs that Brill-Noether loci with $\rho=-1$ have distinct support, and for fixed $r$ give lower bounds on when one direction of the non-containments of the Maximal Brill-Noether Loci Conjecture hold for Brill-Noether loci of rank $r$ linear systems. Using these techniques, we also show that Brill-Noether loci corresponding to rank $2$ linear systems are maximal as soon as $g\ge 28$ and prove the Maximal Brill-Noether Loci Conjecture for $g=20$.

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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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arXiv2023

The embedding theorem in Hurwitz-Brill-Noether Theory

We generalize the Embedding Theorem of Eisenbud-Harris from classical Brill-Noether theory to the setting of Hurwitz-Brill-Noether theory. More precisely, in classical Brill-Noether theory, the embedding theorem states that a general linear series of degree d and rank r on a general curve of genus g is an embedding if r is at least 3. If \(f \colon C \to \mathbb{P}^1\) is a general cover of degree k, and L is a line bundle on C, recent work of the authors shows that the splitting type of \(f_* L\) provides the appropriate generalization of the pair (r, d) in classical Brill--Noether theory. In the context of Hurwitz-Brill-Noether theory, the condition that r is at least 3 is no longer sufficient to guarantee that a general such linear series is an embedding. We show that the additional condition needed to guarantee that a general linear series |L| is an embedding is that the splitting type of \(f_* L\) has at least three nonnegative parts. This new extra condition reflects the unique geometry of k-gonal curves, which lie on scrolls in \(\mathbb{P}^r\).

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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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