Tautological pairings on moduli spaces of curves
Authors:
Renzo Cavalieri and Stephanie Yang
Journal:
Proc. Amer. Math. Soc. 139 (2011), 5162
MSC (2010):
Primary 14N35
Published electronically:
August 23, 2010
MathSciNet review:
2729070
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Additional Information
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 onedimensional in top degree but sometimes do not satisfy Poincaré duality.
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 [AC]
 Enrico Arbarello and Maurizio Cornalba, Combinatorial and algebrogeometric cohomology classes on the moduli spaces of curves, J. Algebraic Geom. 5 (1996), no. 4, 705749. MR 1486986 (99c:14033)
 [Fa]
 Carel Faber, A conjectural description of the tautological ring of the moduli space of curves, Moduli of curves and abelian varieties Aspects Math., E33, Vieweg, Braunschweig, 1999, pp. 109129. MR 1722541 (2000j:14044)
 [FP1]
 Carel Faber and Rahul Pandharipande, Hodge integrals and GromovWitten theory, Invent. Math. 139 (2000), no. 1, 173199. MR 1728879 (2000m:14057)
 [FP2]
 Carel Faber and Rahul Pandharipande, Logarithmic series and Hodge integrals in the tautological ring, Michigan Math. J. 48 (2000), 215252. With an appendix by Don Zagier; dedicated to William Fulton on the occasion of his 60th birthday. MR 1786488 (2002e:14041)
 [FP3]
 Carel Faber and Rahul Pandharipande, Relative maps and tautological classes, J. Eur. Math. Soc. (JEMS) 7 (2005), no. 1, 1349. MR 2120989 (2005m:14046)
 [Fu]
 William Fulton, Intersection theory, 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.
 [GP]
 Tom Graber and Rahul Pandharipande, Constructions of nontautological classes on moduli spaces of curves, Michigan Math. J. 51 (2003), no. 1, 93109. MR 1960923 (2004e:14043)
 [GV1]
 Tom Graber and Ravi Vakil, On the tautological ring of , Turkish J. Math. 25 (2001), no. 1, 237243. MR 1829089 (2002b:14034)
 [GV2]
 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, 137. MR 2176546 (2006j:14035)
 [HM]
 Joe Harris and Ian Morrison, Moduli of curves, Graduate Texts in Mathematics, vol. 187, SpringerVerlag, New York, 1998. MR 1631825 (99g:14031)
 [HKK$^+$]
 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 Math. Institute, Cambridge, MA, 2003. With a preface by Vafa. MR 2003030 (2004g:14042)
 [Ke]
 Sean Keel, Intersection theory of moduli space of stable pointed curves of genus zero, Trans. Amer. Math. Soc. 330 (1992), no. 2, 545574. MR 1034665 (92f:14003)
 [Ko]
 Joachim Kock, Notes on psi classes, available at http://www.mat.uab.es/~kock/ GW.html.
 [L]
 Eduard Looijenga, On the tautological ring of , Invent. Math. 121 (1995), no. 2, 411419. MR 1346214 (96g:14021)
 [M]
 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. 271328. MR 717614 (85j:14046)
 [W]
 J. H. C. Whitehead, On complexes, Ann. of Math. (2) 41 (1940), 809824. MR 0002545 (2,73d)
 [Y]
 Stephanie Yang, Intersection numbers on . Preprint. Available at http://www. stephanieyang.com/Papers.html.
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Additional Information
Renzo Cavalieri
Affiliation:
Department of Mathematics, Colorado State University, Weber Building, Fort Collins, Colorado 805231874
Email:
renzo@math.colostate.edu
Stephanie Yang
Affiliation:
Institutionen för Matematik, Kungliga Tekniska Högskolan, 100 44 Stockholm, Sweden
Email:
stpyang@math.kth.se
DOI:
http://dx.doi.org/10.1090/S000299392010106196
Received by editor(s):
February 24, 2009
Received by editor(s) in revised form:
April 26, 2009, September 1, 2009, December 1, 2009, and January 29, 2010
Published electronically:
August 23, 2010
Communicated by:
Ted Chinburg
Article copyright:
© Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
