The moduli of singular curves on K3 surfaces.

In this article we consider moduli properties of singular curves on K3 surfaces. Let B g denote the stack of primitively polarized K3 surfaces ( X , L ) of genus g and let T g , k n → B g be the stack parameterizing tuples [ ( f : C → X , L ) ] with f an unramified morphism which is birational onto its image, C a smooth curve of genus p ( g , k ) − n and f ⁎ C ∈ | k L | . We show that the forgetful morphism η : T g , k n → M p ( g , k ) − n is generically finite on at least one component, for all but finitely many values of p ( g , k ) − n . We further study the Brill–Noether theory of those curves parametrized by the image of η , and find a Wahl-type obstruction for a smooth curve with an unordered marking to have a nodal model on a K3 surface in such a way that the marking is the divisor over the nodes. [ABSTRACT FROM AUTHOR]