Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Intersections of $\mathbb{Q}$-divisors
on Kontsevich's moduli space $\overline M_{0,n}(\mathbb P^r,d)$
and enumerative geometry

Author: Rahul Pandharipande
Journal: Trans. Amer. Math. Soc. 351 (1999), 1481-1505
MSC (1991): Primary 14N10, 14H10; Secondary 14E99
MathSciNet review: 1407707
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The theory of $\mathbb Q$-Cartier divisors on the space of $n$-pointed, genus 0, stable maps to projective space is considered. Generators and Picard numbers are computed. A recursive algorithm computing all top intersection products of $\mathbb Q$-divisors is established. As a corollary, an algorithm computing all characteristic numbers of rational curves in $\mathbb P^r$ is proven (including simple tangency conditions). Computations of these characteristic numbers are carried out in many examples. The degree of the 1-cuspidal rational locus in the linear system of degree $d$ plane curves is explicitly evaluated.

References [Enhancements On Off] (What's this?)

  • [Al] V. Alexeev, Moduli Spaces $M_{g,n}(W)$ for Surfaces, in Higher-dimensional Complex Varieties (Trento, 1994), de Gruyter, Berlin, 1996, pp. 1-22. CMP 97:16
  • [A] P. Aluffi, The Enumerative Geometry of Plane Cubics II: Nodal and Cuspidal Cubics, Math. Ann., 289 (1991), 543-572. MR 92f:14055
  • [D-I] P. Di Francesco and C. Itzykson, Quantum Intersection Rings, in The Moduli Space of Curves, R. Dijkgraaf, C. Faber, and G. van der Geer, eds., Birkhäuser, 1995, pp. 81-148. MR 96k:11041a
  • [FP] W. Fulton and R. Pandharipande, Notes on Quantum Cohomology and Stable Maps, in Algebraic Geometry-Santa Cruz, 1995, Proc. Sympos. Pure Math., vol. 62, part 2, Amer. Math. Soc., Provicence, RI, 1997, pp. 45-96. MR 98h:14003
  • [Ke] S. Keel, Intersection Theory of Moduli Spaces of Stable $n$-Pointed Curves of Genus $0$, Trans. AMS, 330 (1992), 545-574. MR 92f:14003
  • [K] M. Kontsevich, Enumeration of Rational Curves Via Torus Actions, in The Moduli Space of Curves, R. Dijkgraaf, C. Faber, and G. van der Geer, eds., Birkhäuser, 1995, pp. 335-368. MR 97d:14077
  • [K-M] M. Kontsevich and Y. Manin, Gromov-Witten Classes, Quantum Cohomology, and Enumerative Geometry, Commun. Math. Phys. 164 (1994) 525-562. MR 95i:14049
  • [K-S] S. Kleiman, R. Speiser, Enumerative Geometry of Nodal Plane Curves, Algebraic Geometry - Sundance 1986, Lecture Notes in Mathematics 1311, Springer-Verlag: Berlin, 1988, pp. 156-196. MR 89k:14095
  • [K-S-X] S. Kleiman, S. Stromme, S. Xambo, Sketch of a Verification of Schubert's Number 5,819,539,753,680 of Twisted Cubics, in Space Curves (Roca di Papa, 1985), Lecture Notes in Mathematics 1266, Springer-Verlag: Berlin, 1987, pp. 156-180. MR 88k:14035
  • [P] R. Pandharipande, The Canonical Class of $\overline {M}_{0,n}(\mathbb P^r,d)$ and Enumerative Geometry, Internat. Math. Res. Notices 1997, no. 4, 173-186. MR 98h:14067
  • [R-T] Y. Ruan and G. Tian, A Mathematical Theory of Quantum Cohomology, J. Diff. Geom 42 (1995), 259-367. MR 96m:58033
  • [Sa] G. Saccheiro, Numeri Caratteristici delle Cubishe Piane Nodali, preprint 1985.
  • [S] H. Schubert, Kalkül der Abzählenden Geometrie, B. G. Teubner: Leipzig, 1879.
  • [Z] H. Zeuthen, Almindelige Egenskaber, Danske Videnkabernes Selskabs Skrifter-Natur og Math, 10 (1873).

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 14N10, 14H10, 14E99

Retrieve articles in all journals with MSC (1991): 14N10, 14H10, 14E99

Additional Information

Rahul Pandharipande
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
Address at time of publication: Department of Mathematics, California Institute of Technology, Pasadena, California 91125

Received by editor(s): March 11, 1996
Additional Notes: Partially supported by an NSF Post-Doctoral Fellowship.
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society