Tautological pairings on moduli spaces of curves

Authors:
Renzo Cavalieri and Stephanie Yang

Journal:
Proc. Amer. Math. Soc. **139** (2011), 51-62

MSC (2010):
Primary 14N35

DOI:
https://doi.org/10.1090/S0002-9939-2010-10619-6

Published electronically:
August 23, 2010

MathSciNet review:
2729070

Full-text PDF Free Access

Abstract | References | Similar Articles | 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 one-dimensional in top degree but sometimes do not satisfy Poincaré duality.

**[AC]**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 (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. 109-129. MR**1722541 (2000j:14044)****[FP1]**Carel Faber and Rahul Pandharipande,*Hodge integrals and Gromov-Witten theory*, Invent. Math.**139**(2000), no. 1, 173-199. MR**1728879 (2000m:14057)****[FP2]**Carel Faber and Rahul 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 (2002e:14041)****[FP3]**Carel Faber and Rahul Pandharipande,*Relative maps and tautological classes*, J. Eur. Math. Soc. (JEMS)**7**(2005), no. 1, 13-49. 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, 93-109. MR**1960923 (2004e:14043)****[GV1]**Tom Graber and Ravi Vakil,*On the tautological ring of*, Turkish J. Math.**25**(2001), no. 1, 237-243. 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, 1-37. MR**2176546 (2006j:14035)****[HM]**Joe Harris and Ian Morrison,*Moduli of curves*, Graduate Texts in Mathematics, vol. 187, Springer-Verlag, 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, 545-574. 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, 411-419. 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. 271-328. MR**717614 (85j:14046)****[W]**J. H. C. Whitehead,*On -complexes*, Ann. of Math. (2)**41**(1940), 809-824. MR**0002545 (2,73d)****[Y]**Stephanie Yang,*Intersection numbers on*. Preprint. Available at http://www. stephanieyang.com/Papers.html.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
14N35

Retrieve articles in all journals with MSC (2010): 14N35

Additional Information

**Renzo Cavalieri**

Affiliation:
Department of Mathematics, Colorado State University, Weber Building, Fort Collins, Colorado 80523-1874

Email:
renzo@math.colostate.edu

**Stephanie Yang**

Affiliation:
Institutionen för Matematik, Kungliga Tekniska Högskolan, 100 44 Stockholm, Sweden

Email:
stpyang@math.kth.se

DOI:
https://doi.org/10.1090/S0002-9939-2010-10619-6

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.