## Duke Mathematical Journal

### Relations in the tautological ring of $\mathcal{M}_g$

Eleny-Nicoleta Ionel

#### Abstract

Using a simple geometric argument, we obtain an infinite family of nontrivial relations in the tautological ring of $\mathcal{M}_g$ (coming, in fact, from relations in the Chow ring of $\overline{\mathcal{M}}_{g,2}$). One immediate consequence of these relations is that the classes $\kappa_1,\ldots,\kappa_{[g/3]}$ generate the tautological ring of $\mathcal{M}_g$, which was conjectured by Faber in [F] and recently proven at the level of cohomology by Morita in [M].

#### Article information

Source
Duke Math. J. Volume 129, Number 1 (2005), 157-186.

Dates
First available in Project Euclid: 15 July 2005

http://projecteuclid.org/euclid.dmj/1121448867

Digital Object Identifier
doi:10.1215/S0012-7094-04-12916-1

Mathematical Reviews number (MathSciNet)
MR2155060

Zentralblatt MATH identifier
1086.14023

Subjects
Primary: 14H10: Families, moduli (algebraic)

#### Citation

Ionel, Eleny-Nicoleta. Relations in the tautological ring of ℳ g . Duke Math. J. 129 (2005), no. 1, 157--186. doi:10.1215/S0012-7094-04-12916-1. http://projecteuclid.org/euclid.dmj/1121448867.

#### References

• C. Faber, A conjectural description of the tautological ring of the moduli space of curves'' in Moduli of Curves and Abelian Varieties: The Dutch Intercity Seminar on Moduli, Aspects Math. E33, Vieweg, Braunschweig, Germany, 1999, 109--129.
• T. Graber and R. Vakil, Relative virtual localization and vanishing of tautological classes on moduli spaces of curves, to appear in Duke Math. J., preprint, math.AG/0309227
• J. Harer, Stability of the homology of the mapping class groups of orientable surfaces, Ann. of Math. (2) 121 (1985), 215--249.
• E. Ionel, Topological recursive relations in $H^2g(\mathcalM_g,n)$, Invent. Math. 148 (2002), 627--658.
• E. Looijenga, On the tautological ring of $\mathcalM_g$, Invent. Math. 121 (1995), 411--419.
• S. Morita, Generators for the tautological algebra of the moduli space of curves, Topology 42 (2003), 787--819.
• D. Mumford, Towards an enumerative geometry of the moduli space of curves'' in Arithmetic and Geometry, Vol. II: Geometry: Papers Dedicated to I. R. Shafarevich on the Occasion of His Sixtieth Birthday, Progr. Math. 36, Birkhäuser, Boston, 1983.
• Z. Ran, Curvilinear enumerative geometry, Acta Math. 155 (1985), 81--101.