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

By G. Farkas, Mihnea Cristian Popa. 2003

138 citations recorded by the source corpus.

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).

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c9e18126225aaba8bb54c5cdbfa9f4e3db9b8a61

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. David Jensen
2. A. Verra
3. Sam Payne
4. G. Farkas
5. Eric Katz
6. Dawei Chen
7. Martin Moeller
8. Walter Gubler

In Computational Theory and Mathematics

1. David Jensen
2. Sam Payne
3. A. Verra
4. G. Farkas
5. Dhruv Ranganathan
6. Matthew Baker
7. Eric Katz
8. Walter Gubler

In Geometry and Topology

1. David Jensen
2. A. Verra
3. G. Farkas
4. Sam Payne
5. Dhruv Ranganathan
6. Matthew Baker
7. Eric Katz
8. Dawei Chen

Citations in the corpus

• Moduli of theta-characteristics via Nikulin surfaces (2011, background): “The variety Kg viewed as a subvariety of Mg serves as an obstruction for effective divisors on Mg to having small slope [12] and thus plays a significant role in determining the cone of effective divisors onMg .”
• Tropical independence II: The maximal rank conjecture for quadrics (2015, methodology): “Furthermore, the stabilizations of the flag curves used by Eisenbud and Harris are limits of such curves [Farkas and Popa 2005, Proposition 7.2].”