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Algorithms for computing intersection numbers on moduli spaces of curves, with an application to the class of the locus of Jacobians. (English) Zbl 0952.14042

Hulek, Klaus (ed.) et al., New trends in algebraic geometry. Selected papers presented at the Euro conference, Warwick, UK, July 1996. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 264, 93-109 (1999).
The author describes an algorithm for computing intersection numbers of divisors on the moduli spaces \(\overline{\mathcal M}_{g,n}\) of stable \(n\)-pointed curves of genus \(g.\) The algorithm is based on results of E. Witten [Proc. Conf., Cambridge 1990, Surv. Differ. Geom., Suppl. J. Differ. Geom. 1, 243-310 (1991; Zbl 0757.53049)], M. Kontsevich [Commun. Math. Phys. 147, No. 1, 1-23 (1992; Zbl 0756.35081)], and E. Arbarello and M. Cornalba [J. Algebr. Geom. 5, No. 4, 705-749 (1996; Zbl 0886.14007)]. The author discusses his implementations of the algorithm and the results computed with them. Also, he gives a method for calculating the class of locus of Jacobians in the moduli space of principally polarized abelian varieties of dimension \(g\) and carries this out for \(g\leq 7.\)
Copies of the programs and some tables of intersection numbers computed with them are available from the author: cffaber@math.okstate.edu.
For the entire collection see [Zbl 0913.00032].

MSC:

14Q05 Computational aspects of algebraic curves
14H10 Families, moduli of curves (algebraic)
14-04 Software, source code, etc. for problems pertaining to algebraic geometry
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14H40 Jacobians, Prym varieties
14D20 Algebraic moduli problems, moduli of vector bundles
14K10 Algebraic moduli of abelian varieties, classification

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