Linear sections of the Severi variety and moduli of curves

TL;DR

This paper investigates the geometry of Severi varieties of plane curves, computes invariants of associated stable curves, and derives formulas to understand the moduli space of curves, with applications to divisor slopes and enumerative geometry.

Contribution

It introduces a recursive formula for the degrees of the Hodge bundle on families of curves from Severi varieties, advancing the understanding of moduli space geometry.

Findings

Derived recursive formula for Hodge bundle degrees

Established lower bounds for slopes of divisors on ar{M}_g

Provided applications to enumerative problems on Severi varieties

Abstract

We study the Severi variety $V_{d,g}$ of plane curves of degree $d$ and geometric genus $g$. Corresponding to every such variety, there is a one-parameter family of genus $g$ stable curves whose numerical invariants we compute. Building on the work of Caporaso and Harris, we derive a recursive formula for the degrees of the Hodge bundle on the families in question. For $d$ large enough, these families induce moving curves in $\bar{M}_g$. We use this to derive lower bounds for the slopes of effective divisors on $\bar{M}_g$. Another application of our results is to various enumerative problems on $V_{d,g}$.