Numerical criteria for divisors on M̄g to be ample

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Abstract

The moduli space M̄g,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 M̄g,n is ample if and only if it positively intersects the F-curves. In this paper, proving the F-conjecture on M̄g,n is reduced to showing that certain divisors on M̄0,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 M̄g for g24, 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 M̄g is known to be of general type.

Original languageEnglish (US)
Pages (from-to)1227-1248
Number of pages22
JournalCompositio Mathematica
Volume145
Issue number5
DOIs
StatePublished - Sep 2009
Externally publishedYes

ASJC Scopus subject areas

  • Algebra and Number Theory

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