Stable maps and singular curves on K3 surfaces

Abstract

This thesis develops and applies the theory of arbitrary genus stable maps to K3 surfaces. In the first application of the theory, we study the loci of smooth curves of genus p-n admitting n-nodal models on K3 surfaces. With the exception of finitely many values of p-n, we show these loci have components of the maximum possible dimension p-n+19. We also study the Brill-Noether theory of general points in the loci and consider Wahl-type obstructions for a curve to have an n-nodal model on a K3 surface.<br /> As a second application, we apply the theory of stable maps to a conjecture of Bloch-Beilinson. Let X be a K3 surface admitting a Nikulin involution; i.e. an involution preserving the symplectic form. A conjecture of Bloch-Beilinson predicts that the involution acts as the identity on the Chow group of points. The moduli of K3 surfaces with Nikulin involution come in essentially three different deformation types; we prove the conjecture in one of these three cases.