IVHS of nodal plane curves

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 $π$ has maximal variation if and only if $π= 0$.

💡 Research Summary

This paper, titled “IVHS of nodal plane curves” by Edoardo Sernesi, investigates the infinitesimal variation of Hodge structure (IVHS) for families of irreducible plane curves with nodal singularities. The central objects of study are the Severi varieties V_d,n, which parametrize irreducible plane curves of degree d ≥ 4 having exactly n nodes, where 0 ≤ n ≤ (d-1 choose 2) - 1.

The main theorem (Theorem 1.5) states that for every point