kgonal loci in Severi varieties of curves on K3 surfaces
Andreas Knutsen (U Bergen, Norwegen)  Seminar Algebraische Geometry (SAG)
What 


When 
Jul 04, 2013 from 10:30 am to 11:30 am 
Where  Bonn, Seminarraum MPI, Vivatsgasse 7 
Contact Name  Sachinidis 
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This is a report on joint work with C. Ciliberto. Let $(S,H)$ be a general primitively polarized complex K3 surface $(S,H)$ of genus p. It is wellknown that the Severi varieties of $\delta$nodal curves in $H$, for $0 \leq \delta \leq p$, are smooth and nonempty of dimension $p\delta$. We consider the subloci of curves with kgonal normalizations and prove necessary and sufficient conditions in terms of $p$, $\delta$ and $k$ for these to be nonempty. In contrast to the case of smooth curves, the Severi varieties contain proper subloci of gonalities lower than the maximal gonality given by BrillNoether theory. Besides its intrinsic interest for BrillNoether theory and moduli problems, the subject is related to Mori theory of the 2kdimensional hyperk\"ahler manifold $Hilb^k(S)$ parametrizing 0dimensional length ksubschemes of S, since curves with kgonal normalizations on S give rise to rational curves in $Hilb^k(S)$. I will discuss connections with recent works of Bayer and Macri and talk about some of our expectations.