# Trigonal deformations of rank one and Jacobians

## TL;DR

This paper investigates special infinitesimal deformations of trigonal curves that preserve certain structures, revealing that for high genus or general conditions, the deformation space is very limited, and it clarifies relations between Jacobians and hyperelliptic loci.

## Contribution

It characterizes the infinitesimal deformations of trigonal curves with rank 1 IVHS, completing previous results and identifying the hyperelliptic locus as a unique sub-locus with specific Jacobian domination properties.

## Key findings

- Deformation locus is zero dimensional for genus ≥8 or genus 6,7 with Maroni generality.
- The hyperelliptic locus is the only 2g-1-dimensional sub-locus with Jacobian domination.
- Confirmed and extended results of Naranjo and Pirola regarding Jacobian structures.

## Abstract

In this paper we study the infinitesimal deformations of a trigonal curve that preserve the trigonal series and such that the associate infinitesimal variation of Hodge structure (IVHS) is of rank 1. We show that if the genus g is greater or equal to 8 or g=6,7 and the curve is Maroni general, this locus is zero dimensional. Moreover, we complete a result of Naranjo and Pirola. We show in fact that if the genus g is greater or equal to 6, the hyperelliptic locus is the only 2g-1-dimensional sub-locus Y of the moduli space of curves of genus g, such that for the general element [C] in Y, its Jacobian J(C) is dominated by a hyperelliptic Jacobian of genus g' greater or equal to g.