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Numerical criteria for divisors on $\overline {M}_{g}$ to be ample

Part of: Birational geometry Curves

Published online by Cambridge University Press:  04 September 2009

Angela Gibney
Angela Gibney*
Affiliation:
Department of Mathematics, University of Georgia, Athens, GA 30602, USA (email: agibney@math.uga.edu)
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Abstract

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The moduli space $\M _{g,n}$ of n-pointed stable curves of genus g is stratified by the topological type of the curves being parameterized: the closure of the locus of curves with k nodes has codimension k. The one-dimensional components of this stratification are smooth rational curves called F-curves. These are believed to determine all ample divisors. F-conjecture 

A divisor on $\M _{g,n}$ is ample if and only if it positively intersects theF-curves.

In this paper, proving the F-conjecture on $\M _{g,n}$ is reduced to showing that certain divisors on $\M _{0,N}$ for N⩽g+n are equivalent to the sum of the canonical divisor plus an effective divisor supported on the boundary. Numerical criteria and an algorithm are given to check whether a divisor is ample. By using a computer program called the Nef Wizard, written by Daniel Krashen, one can verify the conjecture for low genus. This is done on $\M _g$ for g⩽24, more than doubling the number of cases for which the conjecture is known to hold and showing that it is true for the first genera such that $\M _g$ is known to be of general type.

MSC classification

Secondary: 14H10: Families, moduli (algebraic) 14E30: Minimal model program (Mori theory, extremal rays)

Information

Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 5 , September 2009 , pp. 1227 - 1248
Copyright
Copyright © Foundation Compositio Mathematica 2009

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