The Prym-canonical Clifford index

Martina Miseri (Università Roma Tre)

Tuesday 28th January 15:00-16:00 Maths 311B

Abstract

The classification problem has always occupied a central role in algebraic geometry. Speaking about algebraic curves, an invariant strictly related to the gonality, that is the minimum degree of a map from a curve C in P^1, is the Clifford index, measuring how many independent global sections a line bundle L on C has for its degree.  After a few recalls on this classic invariant, we give the definition of a new Clifford index computed with respect to the Prym-canonical bundle \omega_C \otimes \eta, where \omega_C is the canonical bundle on C and \eta is a nontrivial 2-torsion line bundle on C. In particular, we will focus on the case of hyperelliptic curves, i.e. curves with a double cover of P^1,  from the study of which is particularly evident that this new Clifford index keeps track of the geometry of the curve C. Specifically, we will show that both its Prym-canonical Clifford index and the surface containing C (that is a rational normal scroll) depend on \eta chosen on C. It will follow that \omega_C \otimes \eta is not normally generated.

During this seminar, we will work over the complex field and C will always be smooth and projective.

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