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

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

2003-05-07

arxiv.org/abs/math/0305112v3

Mathematics

Algebraic Geometry

This version combines our earlier preprints "Effective divisors on M_g and a counterexample to the Slope Conjecture" and "The

Scientific paper

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).

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