Acta Mathematica

The κ ring of the moduli of curves of compact type

Rahul Pandharipande

Abstract

The subalgebra of the tautological ring of the moduli of curves of compact type generated by the κ classes is studied in all genera. Relations, constructed via the virtual geometry of the moduli of stable quotients, are used to obtain minimal sets of generators. Bases and Betti numbers of the κ rings are computed. A universality property relating the higher genus κ rings to the genus 0 rings is proven using the virtual geometry of the moduli space of stable maps. The λg-formula for Hodge integrals arises as the simplest consequence.

Article information

Source
Acta Math., Volume 208, Number 2 (2012), 335-388.

Dates
Received: 24 January 2010
Revised: 15 July 2011
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485892624

Digital Object Identifier
doi:10.1007/s11511-012-0078-2

Mathematical Reviews number (MathSciNet)
MR2931383

Zentralblatt MATH identifier
1273.14057

Rights
2012 © Institut Mittag-Leffler

Citation

Pandharipande, Rahul. The κ ring of the moduli of curves of compact type. Acta Math. 208 (2012), no. 2, 335--388. doi:10.1007/s11511-012-0078-2. https://projecteuclid.org/euclid.acta/1485892624

References

• A rbarello, E. & C ornalba, M., Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves. J. Algebraic Geom., 5 (1996), 705–749.
• B ehrend, K. & F antechi, B., The intrinsic normal cone. Invent. Math., 128 (1997), 45–88.
• C ox, D. A. & K atz, S., Mirror Symmetry and Algebraic Geometry. Mathematical Surveys and Monographs, 68. Amer. Math. Soc., Providence, RI, 1999.
• F aber, C., A conjectural description of the tautological ring of the moduli space of curves, in Moduli of Curves and Abelian Varieties, Aspects Math., E33, pp. 109–129. Vieweg, Braunschweig, 1999.
• F aber, C. & P andharipande, R., Hodge integrals and Gromov–Witten theory. Invent. Math., 139 (2000), 173–199.
• — Logarithmic series and Hodge integrals in the tautological ring. Michigan Math. J., 48 (2000), 215–252.
• — Hodge integrals, partition matrices, and the λg conjecture. Ann. of Math., 157 (2003), 97–124.
• F ulton, W. & P andharipande, R., Notes on stable maps and quantum cohomology, in Algebraic Geometry (Santa Cruz, 1995), Proc. Sympos. Pure Math., 62, pp. 45–96. Amer. Math. Soc., Providence, RI, 1997.
• van der G eer, G., Cycles on the moduli space of abelian varieties, in Moduli of Curves and Abelian Varieties, Aspects Math., E33, pp. 65–89. Vieweg, Braunschweig, 1999.
• G etzler, E., Intersection theory on ${{\overline{\mathcal{M}}}_{1,4 }}$ and elliptic Gromov-Witten invariants. J. Amer. Math. Soc., 10 (1997), 973–998.
• G etzler, E. & P andharipande, R., Virasoro constraints and the Chern classes of the Hodge bundle. Nuclear Phys. B, 530 (1998), 701–714.
• G raber, T. & P andharipande, R., Localization of virtual classes. Invent. Math., 135 (1999), 487–518.
• — Constructions of nontautological classes on moduli spaces of curves. Michigan Math. J., 51 (2003), 93–109.
• G raber, T. & V akil, R., Relative virtual localization and vanishing of tautological classes on moduli spaces of curves. Duke Math. J., 130 (2005), 1–37.
• H assett, B., Moduli spaces of weighted pointed stable curves. Adv. Math., 173 (2003), 316–352.
• I onel, E.-N., Relations in the tautological ring of ${{\mathcal{M}}_g}$. Duke Math. J., 129 (2005), 157–186.
• K ontsevich, M., Enumeration of rational curves via torus actions, in The Moduli Space of Curves (Texel Island, 1994), Progr. Math., 129, pp. 335–368. Birkhäuser, Boston, MA, 1995.
• L osev, A. & M anin, Y., New moduli spaces of pointed curves and pencils of flat connections. Michigan Math. J., 48 (2000), 443–472.
• M arian, A. & O prea, D., Virtual intersections on the Quot scheme and Vafa-Intriligator formulas. Duke Math. J., 136 (2007), 81–113.
• M arian, A., O prea, D. & P andharipande, R., The moduli space of stable quotients. Geom. Topol., 15 (2011), 1651–1706.
• M orita, S., Generators for the tautological algebra of the moduli space of curves. Topology, 42 (2003), 787–819.
• M umford, D., Towards an enumerative geometry of the moduli space of curves, in Arithmetic and Geometry, Vol. II, Progr. Math., 36, pp. 271–328. Birkhäuser, Boston, MA, 1983.
• P andharipande, R., A geometric construction of Getzler’s elliptic relation. Math. Ann., 313 (1999), 715–729.
• — Three questions in Gromov–Witten theory, in Proceedings of the International Congress of Mathematicians (Beijing, 2002), Vol. II, pp. 503–512. Higher Ed. Press, Beijing, 2002.