Cohomology of \({\mathcal M}_ 3\) and \({\mathcal M}^ 1_ 3\). (English) Zbl 0814.14029

Looijenga, Eduard

Bödigheimer, Carl-Friedrich (ed.) et al., Mapping class groups and moduli spaces of Riemann surfaces. Proceedings of workshops held June 24-28, 1991, in Göttingen, Germany, and August 6-10, 1991, in Seattle, WA (USA). Providence, RI: American Mathematical Society. Contemp. Math. 150, 205-228 (1993).

Understanding that the moduli space of smooth non-hyperelliptic genus three curves coincides with that of smooth quartic curves in \(\mathbb{P}^ 2\) and with that of Del Pezzo surfaces of degree 2, the author introduces a stratification into the moduli spaces \(M_ 3\) and \(M^ 1_ 3\) of smooth (pointed) genus three curves according to the configurations of seven points on \(\mathbb{P}^ 2\). From this point of view, he describes the moduli space in terms of a Weyl group and decides the Poincaré polynomials of these strata.
For the entire collection see [Zbl 0777.00025].

Reviewer: T.Sekiguchi (Tokyo)

MSC:

14H10	Families, moduli of curves (algebraic)
14J25	Special surfaces
13D40	Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
14D20	Algebraic moduli problems, moduli of vector bundles
32G15	Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)

Keywords:

stratification of moduli space; moduli space of smooth non-hyperelliptic genus three curves; Del Pezzo surfaces; Poincaré polynomials