SINGULAR CURVES ON A K3 SURFACE AND LINEAR SERIES ON THEIR NORMALIZATIONS

Abstract

We study the Brill–Noether theory of the normalizations of singular, irreducible curves on a K3 surface. We introduce a singular Brill–Noether number ρsing and show that if Pic (K3) = ℤ[L], there are no [Formula: see text]'s on the normalizations of irreducible curves in |L|, provided that ρsing < 0. We give examples showing the sharpness of this result. We then focus on the case of hyperelliptic normalizations, and classify linear systems |L| containing irreducible nodal curves with hyperelliptic normalizations, for ρsing < 0, without any assumption on the Picard group.