Effective divisors on \bar{M}_g, curves on K3 surfaces and the Slope Conjecture

Abstract

We carry out a detailed intersection theoretic analysis of the Deligne-Mumford compactification of the divisor on M_{10} consisting of curves sitting on K3 surfaces. This divisor is not of classical Brill-Noether type, and is known to give a counterexample to the Slope Conjecture. The computation relies on the fact that this divisor has four different incarnations as a geometric subvariety of the moduli space of curves, one of them as a higher rank Brill-Noether divisor consisting of curves with an exceptional rank 2 vector bundle. As an application we describe the birational nature of the moduli space of n-pointed curves of genus 10, for all n. We also show that on M_{11} there are effective divisors of minimal slope and having large Iitaka dimension. This seems to contradict the belief that on M_g the classical Brill-Noether divisors are essentially the only ones of slope 6+12/(g+1).

arXiv commentThis version combines our earlier preprints "Effective divisors on M_g and a counterexample to the Slope Conjecture" and "The geometry of the divisor of K3 sections" into a single paper. To appear in the Journal of Algebraic Geometry

References in corpus (1)

• Green's canonical syzygy conjecture for generic curves of odd genus

Cited by in corpus (10)

• Tropical independence II: The maximal rank conjecture for quadrics
• Koszul cohomology and applications to moduli
• Generalized theta linear series on moduli spaces of vector bundles on curves
• The geometry of the moduli space of odd spin curves
• Effective divisors on moduli spaces of curves and abelian varieties
• The Kodaira dimensions of ${\overline{\mathcal{M}}}_{22}$ and ${\overline{\mathcal{M}}}_{23}$
• Towards the Bertram-Feinberg-Mukai Conjecture
• Birational geometry of moduli of curves with an $S_3$-cover
• Geometry of A_g and Its Compactifications
• Rational maps between moduli spaces of curves and Gieseker-Petri divisors