Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Secants to the Kummer Variety and the minimal Cohomological Class

José Alejandro Aburto

TL;DR

This work addresses when the existence of curves of $(m{+}2)$-secants to the Kummer variety of an indecomposable principally polarized abelian variety implies that the curve represents $m$ times the minimal cohomology class. Building on and generalizing Debarre’s results, the authors remove the requirement $\,\mathrm{End}(X)\simeq \mathbb{Z}$ and establish a main theorem under natural geometric hypotheses, with a detailed analysis of the $m=2$ quadrisecant case that yields an involution and a Prym-type consequence via Krichever-Grushevsky. They show that such secant configurations can be produced from a finite set of secants (one degenerate and $m-1$ nondegenerate), and they discuss how the quadrisecant setup ties to the Abel-Prym criterion, pointing toward an algebro-geometric proof. Collectively, the results contribute to a stratification viewpoint for the moduli of abelian varieties and deepen the link between secant geometry and minimal cohomology classes. The methods combine theta-function identities, endomorphism analysis, and deformation-theoretic arguments to connect geometric secants with cohomological and structural properties of the ambient variety.

Abstract

We prove that, under certain conditions, the existence of a curve of $(m+2)$-secants to the Kummer variety of an indecomposable principally polarized abelian variety $X$, represents $m$-times the minimal cohomological class in $X$. In the case of $m=2$, we find an involution of such curve which proves that \(X\) is a Prym variety by results of Krichever-Grushevsky. This continues the work of Beauville and Debarre, who asked about the relation between some geometric properties of abelian varieties in a way to obtain a stratification of its moduli space.

Secants to the Kummer Variety and the minimal Cohomological Class

TL;DR

This work addresses when the existence of curves of -secants to the Kummer variety of an indecomposable principally polarized abelian variety implies that the curve represents times the minimal cohomology class. Building on and generalizing Debarre’s results, the authors remove the requirement and establish a main theorem under natural geometric hypotheses, with a detailed analysis of the quadrisecant case that yields an involution and a Prym-type consequence via Krichever-Grushevsky. They show that such secant configurations can be produced from a finite set of secants (one degenerate and nondegenerate), and they discuss how the quadrisecant setup ties to the Abel-Prym criterion, pointing toward an algebro-geometric proof. Collectively, the results contribute to a stratification viewpoint for the moduli of abelian varieties and deepen the link between secant geometry and minimal cohomology classes. The methods combine theta-function identities, endomorphism analysis, and deformation-theoretic arguments to connect geometric secants with cohomological and structural properties of the ambient variety.

Abstract

We prove that, under certain conditions, the existence of a curve of -secants to the Kummer variety of an indecomposable principally polarized abelian variety , represents -times the minimal cohomological class in . In the case of , we find an involution of such curve which proves that is a Prym variety by results of Krichever-Grushevsky. This continues the work of Beauville and Debarre, who asked about the relation between some geometric properties of abelian varieties in a way to obtain a stratification of its moduli space.
Paper Structure (4 sections, 4 theorems, 23 equations)

This paper contains 4 sections, 4 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A curve of $(m+2)$-secants and the minimal class
  4. A curve of quadrisecant planes

Key Result

Lemma 3.1

Let $(X,\lambda)$ be an indecomposable principally polarized abelian variety of dimension $g>m$, let $\Theta$ be a symmetric representative of the polarization $\lambda$. Suppose that $Y=\left\lbrace a_1,\ldots,a_{m+2}\right\rbrace$ is a reduced subscheme of $X$ such that $V_Y$ contains an irreducib for a subscheme $Z$ of $X$, then

Theorems & Definitions (12)

  • Definition 2.1
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 3.3
  • Theorem 4.1
  • proof
  • Remark 4.2
  • Corollary 4.3
  • ...and 2 more