A few remarks on sections of the Picard bundle of family of curves

TL;DR

This paper investigates the sections of the Picard bundle in families of algebraic curves, establishing inequalities and classifications that connect geometric properties with the rank of associated normal functions.

Contribution

It introduces a new numerical inequality relating the rank, divisor support, and modular dimension, and classifies cases of equality for families with dominant modular maps.

Findings

Derived a sharp classification for cases of equality involving theta characteristics and canonical sections.

Established a numerical inequality linking rank, divisor support, and modular dimension.

Applied results to geometric problems involving intersections with quartic and quintic curves.

Abstract

We study sections of the relative Picard bundle of a family of curves of genus $g \geq 2$ through the rank of the associated normal function. Using Griffiths' formula for the infinitesimal invariant and higher Schiffer variations, we establish a numerical inequality relating the rank, the minimal support of a representing divisor and the modular dimension of the family. When the modular map is dominant, we obtain a sharp classification: equality occurs only for multiples of odd theta characteristics or of the canonical section. As applications, we derive geometric consequences for plane curves, obtaining results on intersections with very general quartic curves, in the spirit of the work of Chen-Riedl-Yeong, and with quintic curves.