SEVERI VARIETIES AND SELF-RATIONAL MAPS OF K3 SURFACES

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 parameterizing nodal curves of given genus and degree lying on some K3 surface. We also establish a number of numerical constraints satisfied by such nontrivial rational maps, that is of topological degree > 1.