Hodge locus and Brill-Noether type locus

TL;DR

This paper compares the Hodge locus and Brill-Noether type locus in families of smooth projective varieties, establishing relationships between Hodge classes and deformations of line bundles with sections, with applications to curves on surfaces.

Contribution

It introduces a comparison between the Hodge locus and Brill-Noether type locus, providing new insights into their relationship in families of varieties.

Findings

The Hodge locus and Brill-Noether locus are closely related in the context of deformations.

A new method to compare these loci in families of varieties is developed.

Application to curves on surfaces passing through fixed points demonstrates practical relevance.

Abstract

Given a family $\pi:\mc{X} \rightarrow B$ of smooth projective varieties, a closed fiber $\mc{X}_o$ and an invertible sheaf $\mc{L}$ on $\mc{X}_o$, we compare the Hodge locus in $B$ corresponding to the Hodge class $c_1(\mc{L})$ with the locus of points $b\,\in\, B$ such that $\mc{L}$ deforms to an invertible sheaf $\mc{L}_b$ on $\mc{X}_b$ with at least $h^0(\mc{L})$--dimensional space of global sections (it is a Brill-Noether type locus associated to $\mc{L}$). We finally give an application by comparing the Brill-Noether locus to a family of curves on a surface passing through a fixed set of points.