no. 2
Cohomology of the moduli space of non-hyperelliptic genus four curves
[Cohomologie de l’espace de modules des courbes non hyperelliptique de genre quatre]
Fortuna, Mauro1
1 Institut für Algebraische Geometrie Leibniz Universität Hannover Welfengarten 1 30167 Hannover, Germany
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.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3409
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

Fortuna, Mauro 1

1 Institut für Algebraische Geometrie Leibniz Universität Hannover Welfengarten 1 30167 Hannover, Germany
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{AIF_2021__71_2_757_0,
     author = {Fortuna, Mauro},
     title = {Cohomology of the moduli space of non-hyperelliptic genus four curves},
     journal = {Annales de l'Institut Fourier},
     pages = {757--797},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {71},
     number = {2},
     year = {2021},
     doi = {10.5802/aif.3409},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3409/}
}
TY  - JOUR
AU  - Fortuna, Mauro
TI  - Cohomology of the moduli space of non-hyperelliptic genus four curves
JO  - Annales de l'Institut Fourier
PY  - 2021
SP  - 757
EP  - 797
VL  - 71
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3409/
DO  - 10.5802/aif.3409
LA  - en
ID  - AIF_2021__71_2_757_0
ER  - 
%0 Journal Article
%A Fortuna, Mauro
%T Cohomology of the moduli space of non-hyperelliptic genus four curves
%J Annales de l'Institut Fourier
%D 2021
%P 757-797
%V 71
%N 2
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3409/
%R 10.5802/aif.3409
%G en
%F AIF_2021__71_2_757_0
Fortuna, Mauro. Cohomology of the moduli space of non-hyperelliptic genus four curves. Annales de l'Institut Fourier, Tome 71 (2021) no. 2, pp. 757-797. doi : 10.5802/aif.3409. https://aif.centre-mersenne.org/articles/10.5802/aif.3409/

[1] Atiyah, Michael F.; Bott, Raoul The Yang-Mills equations over Riemann surfaces, Philos. Trans. R. Soc. Lond., Volume 308 (1983) no. 1505, pp. 523-615 | MR | Zbl

[2] Beĭlinson, Alexander A.; Bernstein, Joseph; Deligne, Pierre Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) (Astérisque), Volume 100, Société Mathématique de France, 1982, pp. 5-171 | MR | Zbl

[3] Bergström, Jonas; Tommasi, Orsola The rational cohomology of M 4 ¯, Math. Ann., Volume 338 (2007) no. 1, pp. 207-239 | Zbl

[4] Casalaina-Martin, Sebastian; Grushevsky, Samuel; Hulek, Klaus; Laza, Radu Cohomology of the moduli space of cubic threefolds (2020) (https://arxiv.org/abs/1904.08728)

[5] Casalaina-Martin, Sebastian; Jensen, David; Laza, Radu The geometry of the ball quotient model of the moduli space of genus four curves, Compact moduli spaces and vector bundles (Contemporary Mathematics), Volume 564, American Mathematical Society, 2012, pp. 107-136 | DOI | MR | Zbl

[6] Casalaina-Martin, Sebastian; Jensen, David; Laza, Radu Log canonical models and variation of GIT for genus 4 canonical curves, J. Algebr. Geom., Volume 23 (2014) no. 4, pp. 727-764 | DOI | MR

[7] Faber, Carel C.; Pandharipande, Rahul Tautological and non-tautological cohomology of the moduli space of curves, Handbook of moduli. Volume I (Advanced Lectures in Mathematics), Volume 24, International Press; Higher Education Press, 2015, pp. 293-330 | Zbl

[8] Fedorchuk, Maksym The final log canonical model of the moduli space of stable curves of genus 4, Int. Math. Res. Not. (2012) no. 24, pp. 5650-5672 | DOI | MR | Zbl

[9] Fulton, William Intersection theory, Results in Mathematics and Related Areas, 2, Springer, 1998 | DOI | Zbl

[10] Griffiths, Philip A.; Harris, Joseph Principles of algebraic geometry, Pure and Applied Mathematics, John Wiley & Sons, 1978 | Zbl

[11] Hassett, Brendan Classical and minimal models of the moduli space of curves of genus two., Geometric methods in algebra and number theory (Progress in Mathematics), Volume 235, Birkhäuser, 2005, pp. 169-192 | DOI | MR | Zbl

[12] Hassett, Brendan; Hyeon, Donghoon Log canonical models for the moduli space of curves: the first divisorial contraction., Trans. Am. Math. Soc., Volume 361 (2009) no. 8, pp. 4471-4489 | DOI | MR | Zbl

[13] Hassett, Brendan; Hyeon, Donghoon Log minimal model program for the moduli space of stable curves: the first flip, Ann. Math., Volume 177 (2013) no. 3, pp. 911-968 | DOI | MR | Zbl

[14] Hyeon, Donghoon; Lee, Yongnam A birational contraction of genus 2 tails in the moduli space of genus 4 curves I., Int. Math. Res. Not., Volume 2014 (2014) no. 13, pp. 3735-3757 | DOI | MR | Zbl

[15] Kirwan, Frances C. Cohomology of quotients in symplectic and algebraic geometry, Mathematical Notes, 31, Princeton University Press, 1984 | MR | Zbl

[16] Kirwan, Frances C. Partial desingularisations of quotients of nonsingular varieties and their Betti numbers, Ann. Math., Volume 122 (1985) no. 1, pp. 41-85 | DOI | MR | Zbl

[17] Kirwan, Frances C. Rational intersection cohomology of quotient varieties, Invent. Math., Volume 86 (1986) no. 3, pp. 471-505 | DOI | MR | Zbl

[18] Kirwan, Frances C. Moduli spaces of degree d hypersurfaces in P n , Duke Math. J., Volume 58 (1989) no. 1, pp. 39-78 | MR | Zbl

[19] Kirwan, Frances C.; Lee, Ronnie The cohomology of moduli spaces of K3 surfaces of degree 2. I, Topology, Volume 28 (1989) no. 4, pp. 495-516 | DOI | MR | Zbl

[20] Kirwan, Frances C.; Woolf, Jonathan An introduction to intersection homology theory, Chapman & Hall/CRC, 2006 | DOI | Zbl

[21] Kollár, János Shafarevich maps and plurigenera of algebraic varieties, Invent. Math., Volume 113 (1993) no. 1, pp. 177-215 | DOI | MR | Zbl

[22] Mumford, David Bryant Towards an enumerative geometry of the moduli space of curves., Arithmetic and geometry, Vol. II (Progress in Mathematics), Volume 36, 1983, pp. 271-328 | DOI | MR | Zbl

[23] Mumford, David Bryant; Fogarty, John; Kirwan, Frances C. Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], 34, Springer, 1994 | Zbl

[24] Tommasi, Orsola Rational cohomology of the moduli space of genus 4 curves, Compos. Math., Volume 141 (2005) no. 2, pp. 359-384 | DOI | MR | Zbl

Cité par Sources :