2003/05/07 by Gavril Farkas, Farkas, Gavril, Mihnea Popa +1
Mathematics · #14H10 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #math.AG #msc:14H10
paper · pdf · doi:10.48550/arxiv.math/0305112
This 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
openalex publication_date 2003/05/07 · arxiv created 2004/08/31 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We carry out a detailed intersection theoretic analysis of the Deligne-Mumford compactification of the divisor on M10 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 M11 there are effective divisors of minimal slope and having large Iitaka dimension. This seems to contradict the belief that on Mg the classical Brill-Noether divisors are essentially the only ones of slope 6+12/(g+1).