The fourth tautological group of $\overline{\mathfrak{M}}_{g,n}$ and relations with the cohomology

Polito, Marzia

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 14 (2003), p. 137-168 / Harvested from Biblioteca Digitale Italiana di Matematica

Abstract

We give a complete description of the fourth tautological group of the moduli space of pointed stable curves, $\overline{\mathfrak{M}}_{g,n}$, and prove that for $g \ge 8$ it coincides with the cohomology group with rational coefficients. We further give a conjectural upper bound depending on the genus for the degree of new tautological relations.

Résumé

Si dà una descrizione completa del quarto gruppo tautologico dello spazio di moduli delle curve puntate stabili, $\overline{\mathfrak{M}}_{g,n}$, e si dimostra che per $g \ge 8$ tale gruppo coincide con il gruppo di coomologia a coefficienti razionali. Si formula inoltre una congettura sulla dimensione massima del grado delle nuove relazioni tautologiche, in funzione del genere.

Détail

Publié le : 2003-06-01

MR 2053662 | Zbl 1177.14056

Bibliographie

[1] Arbarello, E. - Cornalba, M., Calculating cohomology groups of moduli spaces of curves via algebraic geometry. Inst. Hautes Etudes Sci. Publ. Math., 88, 1998, 97-127. | MR 1733327 | Zbl 0991.14012

[2] Arbarello, E. - Cornalba, M., Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves. J. Algebraic Geometry, 5, 1996, 705-749. | MR 1486986 | Zbl 0886.14007

[3] Arbarello, E. - Cornalba, M. - Griffiths, P. - Harris, J., Geometry of algebraic curves, I. Grundlehren der math. Wiss, vol. 267, Springer-Verlag, New York 1984. | Zbl 0559.14017

[4] Arbarello, E. - Cornalba, M. - Griffiths, P. - Harris, J., Geometry of algebraic curves, II. To appear. | Zbl 0559.14017

[5] Belorousski, P., Chow rings of moduli spaces of pointed elliptic curves. PhD thesis, University of Chicago, 1998. | MR 2716762

[6] Belorousski, P. - Pandharipande, R., A descendent relation in genus $2$. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, vol. XXIX, 2000, 172-191. | MR 1765541 | Zbl 0981.81063

[7] Cornalba, M., Cohomology of Moduli Spaces of Stable Curves. Documenta Mathematica, Extra Vol. ICM 1998, II, 249-257. | MR 1648075 | Zbl 0902.14017

[8] Edidin, D., The codimension-two homology of the moduli space of stable curves is algebraic. Duke Math. Journ., 67, n. 2, 1992, 241-272. | MR 1177306 | Zbl 0766.14017

[9] Faber, C., Chow rings of moduli spaces of curves I: The Chow ring of $\overline{\mathfrak{M}}_{3}$. Annals of Mathematics, 132, 1990, 331-419. | MR 1070600 | Zbl 0721.14013

[10] Faber, C., Chow rings of moduli spaces of curves II: Some result on the Chow ring of $\overline{\mathfrak{M}}_{4}$. Annals of Mathematics, 132, 1990, 421-449. | MR 1078265 | Zbl 0735.14021

[11] Faber, C., Algorithms for computing the intersection numbers on moduli space of curves, with an application to the class of the locus of Jacobians. In: K. Hulek et al. (eds.), New trends in Algebraic Geometry. Cambridge University Press, 1999, 29-45. | MR 1714822 | Zbl 0952.14042

[12] Faber, C., Private communication, 1999.

[13] Faber, C., A conjectural description of the tautological ring of the moduli space of curves. In: C. Faber - E. Looijenga (eds.), Moduli of curves and abelian varieties, The Dutch Intercity Seminar on Moduli. Aspects of Maths., E 33, Vieweg, 1999. | MR 1722541 | Zbl 0978.14029

[14] Getzler, E., Intersection theory on $\overline{\mathfrak{M}}_{1,4}$ and elliptic Gromov-Witten invariants. J. Amer. Math. Soc., 10, n. 4, 1997, 973-998. | MR 1451505 | Zbl 0909.14002

[15] Getzler, E., Topological recursion relations in genus 2. In: M.H. Saito - Y. Shimizu - K. Ueno (eds.), Integrable systems and algebraic geometry (Kobe/Kyoto, 1997). World Sci. Publishing, Singapore-London 1998, 73-106. | MR 1672112 | Zbl 1021.81056

[16] Harer, J., Improved stability for the homology of the mapping class group of orientable surfaces. Duke University Preprint, 1993. | Zbl 0579.57005

[17] Ivanov, N., On the homology stability for Teichmüller modular groups: closed surfaces and twisted coefficients. Contemporary Math., 150, 1993, 149-194. | MR 1234264 | Zbl 0794.32019

[18] Keel, S., Intersection theory of moduli space of stable $n$-pointed curves of genus $0$. Trans. of AMS, 330, n. 2, 1992. | MR 1034665 | Zbl 0768.14002

[19] Loojenga, E., Stable cohomology of the mapping class group with symplectic coefficients and the universal Abel-Jacobi map. J. Algebraic Geometry, 5, 1996, 135-150. | MR 1358038 | Zbl 0860.57010

[20] Mumford, D., Towards an enumerative geometry of the moduli space of curves. In: M. Artin - J. Tate (eds.), Arithmetic and Geometry, vol. II. Progress in Math., 36, Birkhäuser, Boston 1983, 483-510. | MR 717614 | Zbl 0554.14008

[21] Pandharipande, R., A geometric construction of Getzler’s Elliptic relation. Math. Ann., 313, n. 4, 1999, 715-729. | MR 1686935 | Zbl 0933.14035

[22] Polito, M., The fourth cohomology group of the moduli space of stable curves. Tesi di Perfezionamento, Scuola Normale Superiore, Pisa, a.a. 1998-99.

[23] Spanier, E.H., Algebraic Topology. Mc Graw-Hill Series in Higher Math., Mc Graw-Hill, New York-London1996. | MR 210112 | Zbl 0145.43303