Searcharxiv⌕ Search

arXiv subjects

Jesse Leo Kass

Publications and source records attributed to Jesse Leo Kass.

At least 19 recordsLinked to original sources
arXiv2026

Extending the Torelli map to alternative compactifications of the moduli space of curves

Determining the limiting behaviour of the Jacobian as the underlying curve degenerates has been the subject of much interest. For nodal singularities, there are beautiful constructions of Caporaso as well as Pandharipande of compactified universal Jacobians over the moduli space of stable curves. Alexeev later obtained a canonical such compactification by extending the Torelli map out of the Deligne-Mumford compactification of $\mathcal{M}_{g,n}$. In contrast, Alexeev and Brunyate proved that the Torelli map does not extend over the cuspidal locus in Schubert's alternative compactification of pseudostable curves. In this paper, we consider curves with singularities that locally look like the axes in $m$-space, which we call axis-like singularities. We construct an alternative compactification of $\mathcal{M}_{g,n}$ consisting of curves with such singularities and prove that the Torelli map extends out of this compactification. Furthermore, for every alternative compactification in the sense of Smyth, we identify an axis-like locus over which the Torelli map extends.

math.AG↗
arXiv2026

A relative orientation for the moduli space of stable maps to a del Pezzo surface

We prove orientation results for evaluation maps of moduli spaces of rational stable maps to del Pezzo surfaces over a field, both in characteristic $0$ and in positive characteristic. These results and the theory of degree developed in a sequel produce quadratically enriched counts of rational curves over non-algebraically closed fields of characteristic not $2$ or $3$. Orientations are constructed in two steps. First, the ramification locus of the evaluation map is shown to be the divisor in the moduli space of stable maps where image curves have a cusp. Second, this divisor is related to the discriminant of a branched cover of the moduli space given generically by pairs of points on the universal curve with the same image.

math.AG↗
arXiv2026

A quadratically enriched count of rational curves

We define a quadratically enriched count of rational curves in a given divisor class passing through a collection of points on a del Pezzo surface $S$ of degree $\geq 3$ over a perfect field $k$ of characteristic $\neq 2,3.$ When $S$ is $\mathbb{A}^1$-connected, the count takes values in the Grothendieck-Witt group GW(k) of quadratic forms over $k$ and depends only on the divisor class and the fields of definition of the points. More generally, the count is a section of the Grothendieck-Witt sheaf evaluated on $π_0^{\mathbb{A}^1}$ of the restriction of scalars of $S$ corresponding to the fields of definition of the points. We also treat del Pezzo surfaces of degree $2$ under certain conditions. The curve count defined in the present work recovers Gromov-Witten invariants when $k = \mathbb{C}$ and Welschinger invariants when $k = \mathbb{R}.$ To obtain an invariant curve count, we define a quadratically enriched degree for an algebraic map $f$ of $n$-dimensional smooth schemes over a field $k$ under appropriate hypotheses. For example, $f$ can be proper, generically finite and oriented over the complement of a subscheme of codimension $2.$ This degree is compatible with F. Morel's GW(k)-valued degree of an $\mathbb{A}^1$-homotopy class of maps between spheres. For $k \subseteq \mathbb{C}$, this produces an enrichment of the topological degree of a map between manifolds of the same dimension.

math.AG↗
arXiv2024

Joseph Carter Corbin: Arkansas's "Profound Mathematician''

This is a historical article on J. C. Corbin, a nineteenth century mathematician and the founding president of the Historically Black University of Arkansas at Pine Bluff. This version omits the figures that appeared in the published edition. It also includes a lengthier bibliography and a new section ("Added after publication") which addresses some historical points. Updated the section "Added after publication" in response to feedback from Dr. Gladys Turner Finney.

math.HO↗
arXiv2022

What is the height of two points in the plane?

Here we describe the distribution of rational points on the Hilbert scheme of two points in the projective plane. More specifically, we explicitly describe a two-parameter family of height functions $H_{s, t}$, such that the height function associated to any projective embedding is equivalent to some $H_{s, t}$, up to multiplication by a bounded function. For a certain range of the parameters $(s, t)$, we prove an asymptotic formula for the number of rational points of bounded height, and for other $(s, t)$ we obtain an upper bound. The proof establishes an equivalence to a lattice point counting problem, which we solve using the geometry of numbers.

math.NT↗
arXiv2021

The stability space of compactified universal Jacobians

In this paper we describe compactified universal Jacobians, i.e. compactifications of the moduli space of line bundles on smooth curves obtained as moduli spaces of rank 1 torsion-free sheaves on stable curves, using an approach due to Oda-Seshadri. We focus on the combinatorics of the stability conditions used to define compactified universal Jacobians. We explicitly describe an affine space, the stability space, with a decomposition into polytopes such that each polytope corresponds to a proper Deligne-Mumford stack that compactifies the moduli space of line bundles. We apply this description to describe the set of isomorphism classes of compactified universal Jacobians (answering a question of Melo), and to resolve the indeterminacy of the Abel-Jacobi sections (addressing a problem raised by Grushevsky-Zakharov).

math.AG↗
arXiv2021

The class of Eisenbud--Khimshiashvili--Levine is the local A1-Brouwer degree

Given a polynomial function with an isolated zero at the origin, we prove that the local A1-Brouwer degree equals the Eisenbud-Khimshiashvili-Levine class. This answers a question posed by David Eisenbud in 1978. We give an application to counting nodes together with associated arithmetic information by enriching Milnor's equality between the local degree of the gradient and the number of nodes into which a hypersurface singularity degenerates to an equality in the Grothendieck-Witt group.

math.AT↗
arXiv2020

An Arithmetic Count of the Lines on a Smooth Cubic Surface

We give an arithmetic count of the lines on a smooth cubic surface over an arbitrary field $k$, generalizing the counts that over $\mathbb{C}$ there are $27$ lines, and over $\mathbb{R}$ the number of hyperbolic lines minus the number of elliptic lines is $3$. In general, the lines are defined over a field extension $L$ and have an associated arithmetic type $α$ in $L^*/(L^*)^2$. There is an equality in the Grothendieck-Witt group $\operatorname{GW}(k)$ of $k$ $$\sum_{\text{lines}} \operatorname{Tr}_{L/k} \langle α\rangle = 15 \cdot \langle 1 \rangle + 12 \cdot \langle -1 \rangle, $$ where $\operatorname{Tr}_{L/k}$ denotes the trace $\operatorname{GW}(L) \to \operatorname{GW}(k)$. Taking the rank and signature recovers the results over $\mathbb{C}$ and $\mathbb{R}$. To do this, we develop an elementary theory of the Euler number in $\mathbb{A}^1$-homotopy theory for algebraic vector bundles. We expect that further arithmetic counts generalizing enumerative results in complex and real algebraic geometry can be obtained with similar methods.

math.AG↗
arXiv2020

An Example of Wild Ramification in an Enriched Riemann-Hurwitz Formula

M. Levine proved an enrichment of the classical Riemann-Hurwitz formula to an equality in the Grothendieck-Witt group of quadratic forms. In its strongest form, Levine's theorem includes a technical hypothesis on ramification relevant in positive characteristic. We consider wild ramification at points whose residue fields are non-separable extensions of the ground field k. We show an analogous Riemann-Hurwitz formula, and consider an example suggested by S. Saito.

math.AG↗
arXiv2019

Extending the Double Ramification Cycle using Jacobians

We prove that the extension of the double ramification cycle defined by the first-named author (using modifications of the stack of stable curves) coincides with that defined by the last-two named authors (using an extended Brill-Noether locus on suitable compactified universal Jacobians). In particular, in the untwisted case we deduce that both of these extensions coincide with that constructed by Li and Graber-Vakil using a virtual fundamental class on a space of rubber maps.

math.AG↗
arXiv2018

An étale realization which does not exist

For a global field, local field, or finite field $k$ with infinite Galois group, we show that there can not exist a functor from the Morel--Voevodsky $\mathbb{A}^1$-homotopy category of schemes over $k$ to a genuine Galois equivariant homotopy category satisfying a list of hypotheses one might expect from a genuine equivariant category and an étale realization functor. For example, these hypotheses are satisfied by genuine $\mathbb{Z}/2$-spaces and the $\mathbb{R}$-realization functor constructed by Morel--Voevodsky. This result does not contradict the existence of étale realization functors to (pro-)spaces, (pro-)spectra or complexes of modules with actions of the absolute Galois group when the endomorphisms of the unit is not enriched in a certain sense. It does restrict enrichments to representation rings of Galois groups.

math.AT↗
arXiv2018

A classical proof that the algebraic homotopy class of a rational function is the residue pairing

Cazanave has identified the algebraic homotopy class of a rational function of $1$ variable with an explicit nondegenerate symmetric bilinear form. Here we show that Hurwitz's proof of a classical result about real rational functions essentially gives an alternative proof of the stable part of Cazanave's result. We also explain how this result can be interpreted in terms of the residue pairing and that this interpretation relates the result to the signature theorem of Eisenbud, Khimshiashvili, and Levine, showing that Cazanave's result answers a question posed by Eisenbud for polynomial functions in $1$ variable. Finally, we announce results answering this question for functions in an arbitrary number of variables.

math.AG↗
arXiv2017

Extensions of the universal theta divisor

The Jacobian varieties of smooth curves fit together to form a family, the universal Jacobian, over the moduli space of smooth marked curves, and the theta divisors of these curves form a divisor in the universal Jacobian. In this paper we describe how to extend these families over the moduli space of stable marked curves (or rather an open subset thereof) using a stability parameter. We then prove a wall-crossing formula describing how the theta divisor varies with the stability parameter. We use that result to analyze a divisor on the moduli space of smooth marked curves that has recently been studied by Grushevsky-Zakharov, Hain and Müller. In particular, we compute the pullback of the theta divisor studied in Alexeev's work on stable abelic varieties and in Caporaso's work on theta divisors of compactified Jacobians.

math.AG↗
arXiv2017

Autoduality holds for a degenerating abelian variety

We prove that certain degenerate abelian varieties, the compactified Jacobian of a nodal curve and a stable quasiabelian variety, satisfy autoduality. We establish this result by proving a comparison theorem that relates the associated family of Picard schemes to the Néron model, a result of independent interest. In our proof, a key fact is that the total space of a suitable family of compactified Jacobians has rational singularities.

math.AG↗
arXiv2017

Two ways to degenerate the Jacobian are the same

A basic technique for studying a family of Jacobian varieties is to extend the family by adding degenerate fibers. Constructing an extension requires a choice of fibers, and one typically chooses to include either degenerate group varieties or degenerate moduli spaces of sheaves. Here we relate these two different approaches when the base of the family is a regular, 1-dimensional scheme such as a smooth curve. Specifically, we provide sufficient conditions for the line bundle locus in a family of compact moduli spaces of pure sheaves to be isomorphic to the Néron model. The result applies to moduli spaces constructed by Eduardo Esteves and Carlos Simpson, extending results of Busonero, Caporaso, Melo, Oda, Seshadri, and Viviani.

math.AG↗
arXiv2015

An explicit semi-factorial compactification of the Néron model

C.~Pépin recently constructed a semi-factorial compactification of the Néron model of an abelian variety using the flattening technique of Raynaud--Gruson. Here we prove that an explicit semi-factorial compactification is a certain moduli space of sheaves --- the family of compactified jacobians.

math.AG↗
arXiv2015

The compactified jacobian can be nonreduced

We prove by explicit example that the compactified jacobian can be nonreduced. The example is a rational space curve of arithmetic genus 4. This answers a question posed by Cyril D'Souza in 1979.

math.AG↗