Tautological pairings on moduli spaces of curves

Cavalieri, Renzo; Yang, Stephanie

doi:10.1090/S0002-9939-2010-10619-6

Tautological pairings on moduli spaces of curves
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by Renzo Cavalieri and Stephanie Yang PDF

Proc. Amer. Math. Soc. 139 (2011), 51-62 Request permission

Abstract:

We discuss analogs of Faber’s conjecture for two nested sequences of partial compactifications of the moduli space of smooth pointed curves. We show that their tautological rings are one-dimensional in top degree but sometimes do not satisfy Poincaré duality.

References

Enrico Arbarello and Maurizio Cornalba, Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves, J. Algebraic Geom. 5 (1996), no. 4, 705–749. MR 1486986
Carel Faber, A conjectural description of the tautological ring of the moduli space of curves, Moduli of curves and abelian varieties, Aspects Math., E33, Friedr. Vieweg, Braunschweig, 1999, pp. 109–129. MR 1722541
C. Faber and R. Pandharipande, Hodge integrals and Gromov-Witten theory, Invent. Math. 139 (2000), no. 1, 173–199. MR 1728879, DOI 10.1007/s002229900028
C. Faber and R. Pandharipande, Logarithmic series and Hodge integrals in the tautological ring, Michigan Math. J. 48 (2000), 215–252. With an appendix by Don Zagier; Dedicated to William Fulton on the occasion of his 60th birthday. MR 1786488, DOI 10.1307/mmj/1030132716
C. Faber and R. Pandharipande, Relative maps and tautological classes, J. Eur. Math. Soc. (JEMS) 7 (2005), no. 1, 13–49. MR 2120989, DOI 10.4171/JEMS/20
William Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. MR 1644323, DOI 10.1007/978-1-4612-1700-8
T. Graber and R. Pandharipande, Constructions of nontautological classes on moduli spaces of curves, Michigan Math. J. 51 (2003), no. 1, 93–109. MR 1960923, DOI 10.1307/mmj/1049832895
Tom Graber and Ravi Vakil, On the tautological ring of $\overline {\scr M}_{g,n}$, Turkish J. Math. 25 (2001), no. 1, 237–243. MR 1829089
Tom Graber and Ravi Vakil, Relative virtual localization and vanishing of tautological classes on moduli spaces of curves, Duke Math. J. 130 (2005), no. 1, 1–37. MR 2176546, DOI 10.1215/S0012-7094-05-13011-3
Joe Harris and Ian Morrison, Moduli of curves, Graduate Texts in Mathematics, vol. 187, Springer-Verlag, New York, 1998. MR 1631825
Kentaro Hori, Sheldon Katz, Albrecht Klemm, Rahul Pandharipande, Richard Thomas, Cumrun Vafa, Ravi Vakil, and Eric Zaslow, Mirror symmetry, Clay Mathematics Monographs, vol. 1, American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2003. With a preface by Vafa. MR 2003030
Sean Keel, Intersection theory of moduli space of stable $n$-pointed curves of genus zero, Trans. Amer. Math. Soc. 330 (1992), no. 2, 545–574. MR 1034665, DOI 10.1090/S0002-9947-1992-1034665-0
Joachim Kock, Notes on psi classes, available at http://www.mat.uab.es/~kock/ GW.html.
Eduard Looijenga, On the tautological ring of ${\scr M}_g$, Invent. Math. 121 (1995), no. 2, 411–419. MR 1346214, DOI 10.1007/BF01884306
David Mumford, Towards an enumerative geometry of the moduli space of curves, Arithmetic and geometry, Vol. II, Progr. Math., vol. 36, Birkhäuser Boston, Boston, MA, 1983, pp. 271–328. MR 717614
J. H. C. Whitehead, On $C^1$-complexes, Ann. of Math. (2) 41 (1940), 809–824. MR 2545, DOI 10.2307/1968861
Stephanie Yang, Intersection numbers on $\overline {\mathcal {M}}_{g,n}$. Preprint. Available at http://www. stephanieyang.com/Papers.html.