We prove that the moduli space R 15 of Prym curves of genus 15 is of general type. To this end we exhibit a virtual divisor D 15 on R 15 as the degeneracy locus of a globalized multiplication map of sections of line bundles. We then proceed to show that this locus is indeed of codimension one and calculate its class. Using this class, we can conclude that K R 15 is big. This complements a 2010 result of Farkas and Ludwig: now the spaces R g are known to be of general type for g 14.

Twists of Mukai bundles and the geometry of the level 3 modular variety over M 8 Gregor Bruns AbstractFor a curve C of genus 6 or 8 and a torsion bundle η of order ℓ we study the vanishing of the space of global sections of the twist E C ⊗ η of the rank two Mukai bundle E C of C. The bundle E C was used in a well-known construction of Mukai which exhibits general canonical curves of low genus as sections of Grassmannians in the Plücker embedding.Globalizing the vanishing condition, we obtain divisors on the moduli spaces R 6,ℓ and R 8,ℓ of pairs [C, η]. First we characterize these divisors by different conditions on linear series on the level curves, afterwards we calculate the divisor classes. As an application, we are able to prove that R 8,3 is of general type.

We study the stability of the normal bundle of canonical genus 8 curves and prove that on a general curve the bundle is stable. The proof rests on Mukai's description of these curves as linear sections of a Grassmannian G(2, 6). This is the next case of a conjecture by M. Aprodu, G. Farkas, and A. Ortega: the general canonical curve of every genus g 7 should have stable normal bundle. We also give some more evidence for this conjecture in higher genus.

For a curve C of genus 6 or 8 and a torsion bundle η of order ℓ we study the vanishing of the space of global sections of the twist E C ⊗ η of the rank two Mukai bundle E C of C. The bundle E C was used in a well-known construction of Mukai which exhibits general canonical curves of low genus as sections of Grassmannians in the Plücker embedding.Globalizing the vanishing condition, we obtain divisors on the moduli spaces R 6,ℓ and R 8,ℓ of pairs [C, η]. First we characterize these divisors by different conditions on linear series on the level curves, afterwards we calculate the divisor classes. As an application, we are able to prove that R 8,3 is of general type.