Geodesic
The fourth tautological group of $\overline{\mathfrak{M}}_{g,n}$ and relations with the cohomology
Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 14 (2003) no. 2, pp. 137-168.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

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.
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. 
@article{RLIN_2003_9_14_2_a3,
     author = {Polito, Marzia},
     title = {The fourth tautological group of $\overline{\mathfrak{M}}_{g,n}$ and relations with the cohomology},
     journal = {Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni},
     pages = {137--168},
     publisher = {mathdoc},
     volume = {Ser. 9, 14},
     number = {2},
     year = {2003},
     zbl = {1177.14056},
     mrnumber = {1733327},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/RLIN_2003_9_14_2_a3/}
}
TY  - JOUR
AU  - Polito, Marzia
TI  - The fourth tautological group of $\overline{\mathfrak{M}}_{g,n}$ and relations with the cohomology
JO  - Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni
PY  - 2003
SP  - 137
EP  - 168
VL  - 14
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/RLIN_2003_9_14_2_a3/
LA  - en
ID  - RLIN_2003_9_14_2_a3
ER  - 
%0 Journal Article
%A Polito, Marzia
%T The fourth tautological group of $\overline{\mathfrak{M}}_{g,n}$ and relations with the cohomology
%J Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni
%D 2003
%P 137-168
%V 14
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/RLIN_2003_9_14_2_a3/
%G en
%F RLIN_2003_9_14_2_a3
Polito, Marzia. The fourth tautological group of $\overline{\mathfrak{M}}_{g,n}$ and relations with the cohomology. Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 14 (2003) no. 2, pp. 137-168. http://geodesic.mathdoc.fr/item/RLIN_2003_9_14_2_a3/

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

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

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

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

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

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

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

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

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

[10] C. Faber, 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. | DOI | MR | Zbl

[11] C. Faber, 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. | fulltext mini-dml | DOI | MR | Zbl

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

[13] C. Faber, 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. | fulltext mini-dml | MR | Zbl

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

[15] E. Getzler, 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. | fulltext mini-dml | MR | Zbl

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

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

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

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

[20] D. Mumford, 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 | Zbl

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

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

[23] E.H. Spanier, Algebraic Topology. Mc Graw-Hill Series in Higher Math., Mc Graw-Hill, New York-London 1996. | MR | Zbl