Wahl maps and extensions of canonical curves and K3 surfaces

TL;DR

This paper establishes a link between the Gauss--Wahl map's corank of canonical curves and their realization as linear sections of special Gorenstein varieties, extending to polarized K3 surfaces and applications to moduli spaces.

Contribution

It provides new criteria connecting the Gauss--Wahl map's corank with the geometric realization of curves and K3 surfaces as linear sections of Gorenstein varieties, and explores implications for moduli space maps.

Findings

Canonical curves with large Gauss--Wahl corank are linear sections of Gorenstein varieties.

K3 surfaces with certain cohomological conditions are also linear sections of these varieties.

The forgetful map from K3 surfaces with curves to the moduli space of curves has fibers described by the Gauss--Wahl map's corank.

Abstract

Let $C$ be a smooth projective curve of genus $g \geq 11$, non-tetragonal, considered in its canonical embedding in $\mathbf{P}^{g-1}$. We prove that $C$ is a linear section of an arithmetically Gorenstein normal variety $Y$ in $\mathbf{P}^{g+r}$, not a cone, with $\dim(Y)=r+2$ and $\omega_Y=\mathcal{O}_Y(-r)$, if the Gauss--Wahl map of $C$ has corank larger or equal than $r+1$. This relies on previous work of Wahl and Arbarello-Bruno-Sernesi; a partial converse is given via a theorem of Lvovski. We derive a similar result for $K3$ surfaces: Let $(S,L)$ be a polarized $K3$ surface of genus $g \geq 11$, non-tetragonal, and considered in its embedding in $|L|^\vee \cong \mathbf{P}^g$. It is a linear section of a variety $Y$ as above if $H^1(T_S \otimes L^\vee)$ has dimension larger or equal than $r$. We give various applications, including one to the following forgetful modular map: Let $\mathcal{K}_g$ be the moduli space of polarized $K3$ surfaces of genus $g$, and $\mathcal{KC}_g$ the space of pairs $(S,C)$ with $C$ a smooth curve on $S$ and $(S,\mathcal{O}_S(C)) \in \mathcal{K}_g$; we consider the map $c_g: (S,C) \in \mathcal{K}_g \mapsto C \in \mathcal{M}_g$. If $g \geq 11$, we show that this map has smooth fibres over the locus of non-tetragonal curves, with fibre-dimension over non-tetragonal $C$ the corank of the Gauss--Wahl map of $C$ minus one.