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

##### 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)