Abelian varieties in Brill--Noether loci

TL;DR

This paper classifies certain subvarieties within Brill--Noether loci of Jacobians of algebraic curves, focusing on those stable under abelian subvarieties, extending previous results in algebraic geometry.

Contribution

It provides a complete classification of curves with specific subvarieties in their Brill--Noether loci, advancing understanding of the geometric structure of Jacobians.

Findings

Classified curves with special subvarieties in Brill--Noether loci

Extended previous results by Abramovich-Harris and Debarre-Fahlaoui

Identified conditions for stability under abelian subvarieties

Abstract

In this paper, improving on results of Abramovich- Harris and Debarre- Fahlaoui we give the full classification of curves C of genus g such that a Brill--Noether locus W^ s_d(C), strictly contained in the jacobian J(C) of C, contains a variety Z stable under translations by the elements of a positive dimensional abelian subvariety A contained in J(C) and such that dim(Z)=d-dim(A)-2s, i.e., the maximum possible for such a Z.