Rational maps from punctual 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 ${\mathrm{CH}}_0(S)$.

Final version. The proof of Theorem 1.1 in the previous version has been simplified following a suggestion of D.-Q. Zhang

References in corpus (3)

• Symplectic involutions of K3 surfaces act trivially on CH_0
• Symplectic automorphisms of K3 surfaces of arbitrary order
• Bloch's conjecture for Catanese and Barlow surfaces

Cited by in corpus (1)

• Some remarks on modified diagonals