Hannah Larson
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View article: FA-modules of holomorphic forms on $\overline{\mathcal{M}}_{g,n}$
FA-modules of holomorphic forms on $\overline{\mathcal{M}}_{g,n}$ Open
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 cat…
View article: Chow Rings of Hurwitz Spaces with Marked Ramification
Chow Rings of Hurwitz Spaces with Marked Ramification Open
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, p…
View article: Global Brill–Noether theory over the Hurwitzspace
Global Brill–Noether theory over the Hurwitzspace Open
Let $C$ be a curve of genus $g$. A fundamental problem in the theory of\nalgebraic curves is to understand maps $C \\to \\mathbb{P}^r$ of specified degree\n$d$. When $C$ is general, the moduli space of such maps is well-understood by\nthe …
View article: The motivic structures $\mathsf{LS}_{12}$ and $\mathsf{S}_{16}$ in the cohomology of moduli spaces of curves
The motivic structures $\mathsf{LS}_{12}$ and $\mathsf{S}_{16}$ in the cohomology of moduli spaces of curves Open
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 $\m…
View article: Complex Bott Periodicity in algebraic geometry
Complex Bott Periodicity in algebraic geometry Open
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 classica…
View article: On the Chow and cohomology rings of moduli spaces of stable curves
On the Chow and cohomology rings of moduli spaces of stable curves Open
In this paper, we ask: For which (g, n) is the rational Chow or cohomology ring of \bar{\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 tauto…
View article: Moduli spaces of curves with polynomial point counts
Moduli spaces of curves with polynomial point counts Open
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 modu…
View article: Brill--Noether theory of smooth curves in the plane and on Hirzebruch surfaces
Brill--Noether theory of smooth curves in the plane and on Hirzebruch surfaces Open
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 …
View article: The Chow ring of the universal Picard stack over the hyperelliptic locus
The Chow ring of the universal Picard stack over the hyperelliptic locus Open
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 \…
Extensions of tautological rings and motivic structures in the cohomology of ${\overline {\mathcal {M}}}_{g,n}$ Open
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 a…
View article: Maximal Brill--Noether loci via the gonality stratification
Maximal Brill--Noether loci via the gonality stratification Open
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 $ρ=-1$ have distinct support, and for fixed $r$ …
View article: Extensions of tautological rings and motivic structures in the cohomology of $\overline{\mathcal{M}}_{g,n}$
Extensions of tautological rings and motivic structures in the cohomology of $\overline{\mathcal{M}}_{g,n}$ Open
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 c…
View article: The embedding theorem in Hurwitz-Brill-Noether Theory
The embedding theorem in Hurwitz-Brill-Noether Theory Open
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 linea…
The eleventh cohomology group of Open
We prove that the rational cohomology group $H^{11}(\overline {\mathcal {M}}_{g,n})$ vanishes unless $g = 1$ and $n \geq 11$ . We show furthermore that $H^k(\overline {\mathcal {M}}_{g,n})$ is pure Hodge–Tate for all even $k \leq 12$ and d…
View article: The bielliptic locus in genus 11
The bielliptic locus in genus 11 Open
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…
View article: The eleventh cohomology group of $\bar{\mathcal{M}}_{g,n}$
The eleventh cohomology group of $\bar{\mathcal{M}}_{g,n}$ Open
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 $\# …
View article: On the Chow and cohomology rings of moduli spaces of stable curves
On the Chow and cohomology rings of moduli spaces of stable curves Open
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 ring…
View article: The rational Chow rings of moduli spaces of hyperelliptic curves with marked points
The rational Chow rings of moduli spaces of hyperelliptic curves with marked points Open
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 $\mathcal{I}_{g,n}$, pa…
View article: The integral Picard groups of low-degree Hurwitz spaces
The integral Picard groups of low-degree Hurwitz spaces Open
We compute the Picard groups with integral coefficients of the Hurwitz stacks parametrizing degree $4$ and $5$ covers of $\mathbb{P}^1$. As a consequence, we also determine the integral Picard groups of the Hurwitz stacks parametrizing sim…
View article: Chow rings of low-degree Hurwitz spaces
Chow rings of low-degree Hurwitz spaces Open
While there is much work and many conjectures surrounding the intersection theory of the moduli space of curves, relatively little is known about the intersection theory of the Hurwitz space $\mathcal{H}_{k, g}$ parametrizing smooth degree…
View article: On an equivalence of divisors on $\bar{M}_{0,n}$ from Gromov-Witten theory and conformal blocks
On an equivalence of divisors on $\bar{M}_{0,n}$ from Gromov-Witten theory and conformal blocks Open
We consider a conjecture that identifies two types of base point free divisors on $\bar{M}_{0,n}$. The first arises from Gromov-Witten theory of a Grassmannian. The second comes from first Chern classes of vector bundles associated to simp…
View article: The intersection theory of the moduli stack of vector bundles on $\mathbb{P}^1$
The intersection theory of the moduli stack of vector bundles on $\mathbb{P}^1$ Open
We determine the integral Chow and cohomology rings of the moduli stack $\mathcal{B}_{r,d}$ of rank $r$, degree $d$ vector bundles on $\mathbb{P}^1$ bundles. We first show that the rational Chow ring $A_{\mathbb{Q}}^*(\mathcal{B}_{r,d})$ i…
View article: The Chow rings of the moduli spaces of curves of genus 7, 8, and 9
The Chow rings of the moduli spaces of curves of genus 7, 8, and 9 Open
The rational Chow ring of the moduli space $\mathcal{M}_g$ of curves of genus $g$ is known for $g \leq 6$. Here, we determine the rational Chow rings of $\mathcal{M}_7, \mathcal{M}_8,$ and $\mathcal{M}_9$ by showing they are tautological. …
View article: Intersection theory on low-degree Hurwitz spaces
Intersection theory on low-degree Hurwitz spaces Open
While there is much work and many conjectures surrounding the intersection theory of the moduli space of curves, relatively little is known about the intersection theory of the Hurwitz space $\mathscr{H}_{k, g}$ parametrizing smooth degree…
View article: Tautological classes on low-degree Hurwitz spaces
Tautological classes on low-degree Hurwitz spaces Open
Let $\mathcal{H}_{k,g}$ be the Hurwitz stack parametrizing degree $k$, genus $g$ covers of $\mathbb{P}^1$. We define the tautological ring of $\mathcal{H}_{k,g}$ and we show that all Chow classes, except possibly those supported on the loc…
View article: Global Brill--Noether Theory over the Hurwitz Space
Global Brill--Noether Theory over the Hurwitz Space Open
Let $C$ be a curve of genus $g$. A fundamental problem in the theory of algebraic curves is to understand maps $C \to \mathbb{P}^r$ of specified degree $d$. When $C$ is general, the moduli space of such maps is well-understood by the main …
View article: Normal Bundles of Lines on Hypersurfaces
Normal Bundles of Lines on Hypersurfaces Open
Let $X \subset \mathbb{P}^n$ be a smooth hypersurface. Given a sequence of integers $\vec{a} = (a_1, \ldots, a_{n-2})$ with $a_1 \leq \cdots \leq a_{n-2}$, let $F_{\vec{a}}(X)$ be the parameter space of lines $L$ on $X$ such that $N_{L/X} …