Singular curves on a K3 surface and linear series on their normalizations

Flamini, Flaminio; Knutsen, Andreas Leopold; Pacienza, Gianluca

Singular curves on a K3 surface and linear series on their normalizations

Authors

Flamini, Flaminio

Knutsen, Andreas Leopold

Pacienza, Gianluca

Description

In this paper, we study the Brill-Noether theory of the normalizations of singular, irreducible curves on a $K3$ surface. We introduce a {\em singular} Brill-Noether number $ρ_{sing}$ and show that if the Picard group of the K3 surface is ${\mathbb Z} [L]$, there are no $g^r_d$'s on the normalizations of irreducible curves in $|L|$, provided that $ρ_{sing} <0$. We give examples showing the sharpness of this result. We then focus on the case of {\em hyperelliptic normalizations}, and classify linear systems $|L|$ containing irreducible nodal curves with hyperelliptic normalizations, for $ρ_{sing}<0$, without any assumption on its Picard group.
20 pages, remarks of the referee are added, to be published on International Journal of Mathematics