Cohomology of the moduli space of non-hyperelliptic genus four curves

Fortuna, Mauro

Annales de l'Institut Fourier, Tome 71 (2021) no. 2, pp. 757-797.

Nous calculons les nombres de Betti d’intersection du modèle GIT de l’espace de modules des courbes générales pour Brill–Noether–Petri du genre quatre. Cet espace s’est révélé être le dernier modèle log canonique non trivial pour l’espace de modules des courbes stables de genre quatre, dans le cadre du programme de Hassett et Keel. La stratégie du calcul cohomologique s’appuie sur une méthode générale développée par Kirwan pour calculer la cohomologie des quotients GIT des variétés projectives, basée sur la stratification équivariante parfaite des points instables étudiés par Hesselink et autres et une résolution partielle des singularités.

We compute the intersection Betti numbers of the GIT model of the moduli space of Brill–Noether–Petri general curves of genus four. This space was shown to be the final non-trivial log canonical model for the moduli space of stable genus four curves, under the Hassett–Keel program. The strategy of the cohomological computation relies on a general method developed by Kirwan to calculate the cohomology of GIT quotients of projective varieties, based on the equivariantly perfect stratification of the unstable points studied by Hesselink and others and a partial resolution of singularities.

Classification : 14L24, 14F43, 14H10
Keywords: Geometric Invariant Theory, moduli of curves, curves of genus four, intersection cohomology.
Mot clés : Théorie géométrique des invariants, modules des courbes, courbes de genre quatre, cohomologie d’intersection

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