Rational maps from punctual Hilbert schemes of K3 surfaces

TL;DR

This paper investigates dominant rational maps from punctual Hilbert schemes of K3 surfaces with many rational curves, showing their images are rationally connected unless the map is generically finite, and applies this to symplectic involutions.

Contribution

It establishes a new property of rational maps from Hilbert schemes of K3 surfaces and simplifies Voisin's proof regarding symplectic involutions.

Findings

Images are rationally connected if the map is not generically finite.

Simplified proof of Voisin's result on symplectic involutions acting trivially on CH_0.

Provides insights into the structure of rational maps from Hilbert schemes of K3 surfaces.

Abstract

The purpose of this short note is to study dominant rational maps from punctual Hilbert schemes of length $k>1$ of projective K3 surfaces $S$ containing infinitely many rational curves. Precisely, we prove that their image is necessarily rationally connected if this rational map is not generically finite. As an application, we simplify the proof of C. Voisin's of the fact that symplectic involutions of any projective K3 surface $S$ act trivially on $\mathrm{CH}_0(S)$.