The tautological ring of $M_{1,n}^{ct}$

Nous décrivons l’anneau tautologique de l’espace des modules des courbes stables de genre un de type compact avec $n$ points marqués. On prouve que c’est une algèbre de Gorenstein.

We describe the tautological ring of the moduli space of stable $n$-pointed curves of genus one of compact type. It is proven that it is a Gorenstein algebra.

[1] Arbarello, E.; Cornalba, M. Calculating cohomology groups of moduli spaces of curves via algebraic geometry, Inst. Hautes Études Sci. Publ. Math. (1998) no. 88, p. 97-127 (1999) | EuDML 104137 | Numdam | MR 1733327 | Zbl 0991.14012

[2] Cornalba, M. On the projectivity of the moduli spaces of curves, J. Reine Angew. Math., Volume 443 (1993), pp. 11-20 | Article | EuDML 153559 | MR 1241126 | Zbl 0781.14017

[3] Faber, C. A conjectural description of the tautological ring of the moduli space of curves, Moduli of curves and abelian varieties (Aspects Math., E33), Vieweg, Braunschweig, 1999, pp. 109-129 | MR 1722541 | Zbl 0978.14029

[4] Faber, C. Hodge integrals, tautological classes and Gromov-Witten theory, Proceedings of the Workshop “Algebraic Geometry and Integrable Systems related to String Theory” (Kyoto, 2000) (2001) no. 1232, pp. 78-87 | MR 1905884

[5] Faber, C. A remark on a conjecture of Hain and Looijenga (2008) (Arxiv preprint arXiv:0812.3631) | Zbl 1278.14037

[6] Faber, C.; Pandharipande, R. Logarithmic series and Hodge integrals in the tautological ring, Michigan Math. J., Volume 48 (2000), pp. 215-252 (With an appendix by Don Zagier, Dedicated to William Fulton on the occasion of his 60th birthday) | Article | MR 1786488 | Zbl 1090.14005

[7] Faber, C.; Pandharipande, R. Hodge integrals, partition matrices, and the ${\lambda }_g$ conjecture, Annals of mathematics (2003), pp. 97-124 | MR 1954265 | Zbl 1058.14046

[8] Faber, C.; Pandharipande, R. Relative maps and tautological classes, J. Eur. Math. Soc. (JEMS), Volume 7 (2005) no. 1, pp. 13-49 | Article | EuDML 277270 | MR 2120989 | Zbl 1084.14054

[9] Fulton, W.; MacPherson, R. A compactification of configuration spaces, Ann. of Math. (2), Volume 139 (1994) no. 1, pp. 183-225 | Article | MR 1259368 | Zbl 0820.14037

[10] Getzler, E. Intersection theory on ${\overline{M}}_{1,4}$ and elliptic Gromov-Witten invariants, Journal of the American Mathematical Society, Volume 10 (1997) no. 4, pp. 973 | MR 1451505 | Zbl 0909.14002

[11] Graber, T.; Pandharipande, R. Constructions of nontautological classes on moduli spaces of curves, Michigan Math. J., Volume 51 (2003) no. 1, pp. 93-109 | Article | MR 1960923 | Zbl 1079.14511

[12] Graber, T.; Vakil, R. Relative virtual localization and vanishing of tautological classes on moduli spaces of curves, Duke Math. J., Volume 130 (2005) no. 1, pp. 1-37 | Article | MR 2176546 | Zbl 1088.14007

[13] Green, M.; Griffiths, P. An interesting 0-cycle, Duke Math. J., Volume 119 (2003) no. 2, pp. 261-313 | Article | MR 1997947 | Zbl 1058.14014

[14] Hain, R.; Looijenga, E. Mapping class groups and moduli spaces of curves, Algebraic geometry—Santa Cruz 1995 (Proc. Sympos. Pure Math.), Volume 62, Amer. Math. Soc., Providence, RI, 1997, pp. 97-142 | MR 1492535 | Zbl 0914.14013

[15] Hanlon, P.; Wales, D. On the decomposition of Brauer’s centralizer algebras, J. Algebra, Volume 121 (1989) no. 2, pp. 409-445 | Article | MR 992775 | Zbl 0695.20026

[16] Keel, S. Intersection theory of moduli space of stable $n$-pointed curves of genus zero, Trans. Amer. Math. Soc., Volume 330 (1992) no. 2, pp. 545-574 | Article | MR 1034665 | Zbl 0768.14002

[17] Pandharipande, R. A geometric construction of Getzler’s elliptic relation, Math. Ann., Volume 313 (1999) no. 4, pp. 715-729 | Article | MR 1686935 | Zbl 0933.14035