Second fundamental form and higher Gaussian maps

TL;DR

This paper explores the relationship between higher Gaussian maps and the second fundamental form of the Torelli map on algebraic curves, establishing new injectivity results and rank estimates for these maps.

Contribution

It generalizes previous results linking Gaussian maps and the Torelli map, providing new injectivity and rank bounds for even Gaussian maps on non-hyperelliptic curves.

Findings

The Gaussian map μ_{6g-6} is injective for non-hyperelliptic curves.

All even Gaussian maps μ_{2k} are zero for k > 3g-3.

Provides estimates for the rank of μ_{2k} for g-1 ≤ k ≤ 3g-3.

Abstract

In this paper we show a relation between higher even Gaussian maps of the canonical bundle on a smooth projective curve of genus $g \geq 4$ and the second fundamental form of the Torelli map. This generalises a result obtained by Colombo, Pirola and Tortora on the second Gaussian map and the second fundamental form. As a consequence, we prove that for any non-hyperelliptic curve, the Gaussian map $\mu_{6g-6}$ is injective, hence all even Gaussian maps $\mu_{2k}$ are identically zero for all $k >3g-3$. We also give an estimate for the rank of $\mu_{2k}$ for $g-1 \leq k \leq 3g-3.$