Severi varieties and self rational maps of K3 surfaces

TL;DR

This paper explores the relationship between self-rational maps of generic K3 surfaces and the irreducibility of Severi varieties, providing numerical constraints on non-trivial maps and contributing to conjectures in algebraic geometry.

Contribution

It establishes a connection between the conjecture on triviality of self-rational maps and Severi varieties' irreducibility, offering new numerical constraints on non-trivial maps.

Findings

Self-rational maps of generic K3 surfaces are conjectured to be trivial.

Numerical constraints are derived for non-trivial rational maps with topological degree >1.

The work links the triviality conjecture to Severi varieties' properties.

Abstract

Self-rational maps of generic algebraic K3 surfaces are conjectured to be trivial. We relate this conjecture to a conjecture concerning the irreducibility of the universal Severi varieties parametrizing nodal curves of given genus and degree lying on some K3 surface. We also establish a number of numerical constraints satisfied by such non trivial rational maps, that is of topological degree >1.