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Numerical criteria for divisors on to be ample

  • Angela Gibney (a1)
Abstract
Abstract

The moduli space 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. <span class='italic'>F</span>-<span class='sc'>conjecture</span> 

A divisor on  is ample if and only if it positively intersects theF-curves.

In this paper, proving the F-conjecture on is reduced to showing that certain divisors on for Ng+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 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 is known to be of general type.

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[1] E. Arbarello and M. Cornalba , Calculating cohomology groups of moduli spaces of curves via algebraic geometry, Publ. Math. Inst. Hautes Études Sci. 88 (1998), 97127.

[3] G. Farkas , The geometry of the moduli space of curves of genus 23, Math. Ann. 318 (2000), 4365.

[4] G. Farkas , Syzygies of curves and the effective cone of $\overline {M}_g$, Duke Math. J. 135 (2006), 5398.

[5] G. Farkas and A. Gibney , The mori cones of moduli spaces of pointed curves of small genus, Trans. Amer. Math. Soc. 355 (2003), 11831199.

[7] A. Gibney , S. Keel and I. Morrison , Towards the ample cone of $\overline {M}_{g,n}$, J. Amer. Math. Soc. (2) 15 (2001), 273294.

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Compositio Mathematica
  • ISSN: 0010-437X
  • EISSN: 1570-5846
  • URL: /core/journals/compositio-mathematica
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