Special divisors on curves on K3 surfaces carrying an Enriques involution.
Marco Ramponi
(Univ. Poitiers - France)

Recent works by Farkas and Kemeny on the Green-Lazarsfeld and Prym-Green conjectures rely on the possibility of computing Clifford indices and gonalities of curves on special K3 surfaces. In this talk, we want to study the pencils of minimal degree on the smooth curves lying on a K3 surface X which carries a fixed-point free involution. Such an automorphism is also called an Enriques involution, since the quotient of X by it is an Enriques surface. Building on work by Knutsen and Lopez around the Brill-Noether theory for curves on Enriques surfaces, we show that, for generic X, the gonality and the Clifford index of all curves on X is governed by the genus 1 fibrations carried by the K3 surface X.