The Universal Severi Variety of Rational Curves on K3 Surfaces
  We investigate the universal Severi variety of rational curves on K3 surfaces, which parametrises irreducible rational curves in a fixed class on varying K3 surfaces of fixed genus. We investigate the conjecuted irreducibility of this space in the case of primitive classes using the deformation theory of stable maps and obtain partial results. In an appendix we give a short proof of the irreducibility of universal Severi varieties of high genus curves on K3 surfaces. 
Michael Kemeny
math.AG