IVHS of nodal plane curves

TL;DR

This paper investigates the variation of Hodge structures in families of nodal plane curves, establishing maximal variation for certain Severi varieties and related curve families, advancing understanding of their geometric properties.

Contribution

It proves maximal infinitesimal variation of Hodge structures for Severi varieties of nodal plane curves and characterizes when families of curves mapping to a fixed curve have maximal variation.

Findings

Maximal variation of Hodge structures for Severi varieties of nodal plane curves.

Family of genus g ≥ 1 curves mapping to a fixed curve Y has maximal variation iff Y has genus 0.

Provides conditions for maximal variation in families of curves with fixed genus and degree.

Abstract

Let $\V_{d,n}$ be the Severi variety of irreducible plane curves of degree $d\ge 4$ having $n$ nodes, with $0\le n \le \binom{d-1}{2}-1$. We prove that for every $[\ol C]\in \V_{d,n}$, the infinitesimal variation of the Hodge structure of the normalization $C$ of $\ol C$ is maximal as $[\ol C]$ moves in $\V_{d,n}$. As a preliminary result, we also prove that the family of curves of genus $g \ge 1$ mapping with degree $d \ge 2$ to a fixed curve $Y$ of genus $\pi$ has maximal variation if and only if $\pi = 0$.