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

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- by Rahul Pandharipande
- Trans. Amer. Math. Soc.
**351**(1999), 1481-1505 - DOI: https://doi.org/10.1090/S0002-9947-99-01909-1
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## 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

- V. Alexeev,
*Moduli Spaces $M_{g,n}(W)$ for Surfaces*, in*Higher-dimensional Complex Varieties*(*Trento, 1994*), de Gruyter, Berlin, 1996, pp. 1–22. - Paolo Aluffi,
*The enumerative geometry of plane cubics. II. Nodal and cuspidal cubics*, Math. Ann.**289**(1991), no. 4, 543–572. MR**1103035**, DOI 10.1007/BF01446588 - Hermann Kober,
*Transformationen von algebraischem Typ*, Ann. of Math. (2)**40**(1939), 549–559 (German). MR**96**, DOI 10.2307/1968939 - János Kollár, Robert Lazarsfeld, and David R. Morrison (eds.),
*Algebraic geometry—Santa Cruz 1995*, Proceedings of Symposia in Pure Mathematics, vol. 62, American Mathematical Society, Providence, RI, 1997. MR**1492532**, DOI 10.1090/pspum/062.2 - 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 - Maxim Kontsevich,
*Enumeration of rational curves via torus actions*, The moduli space of curves (Texel Island, 1994) Progr. Math., vol. 129, Birkhäuser Boston, Boston, MA, 1995, pp. 335–368. MR**1363062**, DOI 10.1007/978-1-4612-4264-2_{1}2 - M. Kontsevich and Yu. Manin,
*Gromov-Witten classes, quantum cohomology, and enumerative geometry*, Comm. Math. Phys.**164**(1994), no. 3, 525–562. MR**1291244**, DOI 10.1007/BF02101490 - Steven L. Kleiman and Robert Speiser,
*Enumerative geometry of nodal plane cubics*, Algebraic geometry (Sundance, UT, 1986) Lecture Notes in Math., vol. 1311, Springer, Berlin, 1988, pp. 156–196. MR**951646**, DOI 10.1007/BFb0082914 - S. Kleiman, S. A. Strømme, and S. Xambó,
*Sketch of a verification of Schubert’s number $5\,819\,539\,783\,680$ of twisted cubics*, Space curves (Rocca di Papa, 1985) Lecture Notes in Math., vol. 1266, Springer, Berlin, 1987, pp. 156–180. MR**908713**, DOI 10.1007/BFb0078183 - Rahul Pandharipande,
*The canonical class of $\overline {M}_{0,n}(\mathbf P^r,d)$ and enumerative geometry*, Internat. Math. Res. Notices**4**(1997), 173–186. MR**1436774**, DOI 10.1155/S1073792897000123 - Yongbin Ruan and Gang Tian,
*A mathematical theory of quantum cohomology*, J. Differential Geom.**42**(1995), no. 2, 259–367. MR**1366548** - G. Saccheiro,
*Numeri Caratteristici delle Cubishe Piane Nodali*, preprint 1985. - H. Schubert,
*Kalkül der Abzählenden Geometrie*, B. G. Teubner: Leipzig, 1879. - H. Zeuthen,
*Almindelige Egenskaber*, Danske Videnkabernes Selskabs Skrifter-Natur og Math, 10 (1873).

## Bibliographic 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
- MR Author ID: 357813
- Email: rahulp@cco.caltech.edu
- Received by editor(s): March 11, 1996
- Additional Notes: Partially supported by an NSF Post-Doctoral Fellowship.
- © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**351**(1999), 1481-1505 - MSC (1991): Primary 14N10, 14H10; Secondary 14E99
- DOI: https://doi.org/10.1090/S0002-9947-99-01909-1
- MathSciNet review: 1407707