The degree of the variety of rational ruled surfaces and Gromov-Witten invariants
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- by Cristina Martínez PDF
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Abstract:
We compute the degree of the variety parametrizing rational ruled surfaces of degree $d$ in $\mathbb {P}^{3}$ by relating the problem to Gromov-Witten invariants and Quantum cohomology.References
- Allen Altman and Steven Kleiman, Introduction to Grothendieck duality theory, Lecture Notes in Mathematics, Vol. 146, Springer-Verlag, Berlin-New York, 1970. MR 0274461
- Paolo Aluffi, Two characteristic numbers for smooth plane curves of any degree, Trans. Amer. Math. Soc. 329 (1992), no. 1, 73–96. MR 1041042, DOI 10.1090/S0002-9947-1992-1041042-5
- Aaron Bertram, Quantum Schubert calculus, Adv. Math. 128 (1997), no. 2, 289–305. MR 1454400, DOI 10.1006/aima.1997.1627
- Aaron Bertram, Towards a Schubert calculus for maps from a Riemann surface to a Grassmannian, Internat. J. Math. 5 (1994), no. 6, 811–825. MR 1298995, DOI 10.1142/S0129167X94000401
- Daniel F. Coray and Israel Vainsencher, Enumerative formulae for ruled cubic surfaces and rational quintic curves, Comment. Math. Helv. 61 (1986), no. 3, 501–518. MR 860137, DOI 10.1007/BF02621930
- David Eisenbud and Joe Harris, On varieties of minimal degree (a centennial account), Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 3–13. MR 927946, DOI 10.1090/pspum/046.1/927946
- P. Di Francesco and C. Itzykson, Quantum intersection rings, The moduli space of curves (Texel Island, 1994) Progr. Math., vol. 129, Birkhäuser Boston, Boston, MA, 1995, pp. 81–148. MR 1363054, DOI 10.1007/978-1-4612-4264-2_{4}
- Robert Friedman, Algebraic surfaces and holomorphic vector bundles, Universitext, Springer-Verlag, New York, 1998. MR 1600388, DOI 10.1007/978-1-4612-1688-9
- W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology, Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 45–96. MR 1492534, DOI 10.1090/pspum/062.2/1492534
- William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR 732620, DOI 10.1007/978-3-662-02421-8
- Alexander Grothendieck, Techniques de construction et théorèmes d’existence en géométrie algébrique. IV. Les schémas de Hilbert, Séminaire Bourbaki, Vol. 6, Soc. Math. France, Paris, 1995, pp. Exp. No. 221, 249–276 (French). MR 1611822
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
- Steven L. Kleiman, The transversality of a general translate, Compositio Math. 28 (1974), 287–297. MR 360616
- Joachim Kock and Israel Vainsencher, A fórmula de Kontsevich para curvas racionais planas, 22$^\textrm {o}$ Colóquio Brasileiro de Matemática. [22nd Brazilian Mathematics Colloquium], Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 1999 (Portuguese). MR 1722053
- M. Kontsevich and Yu. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994), no. 3, 525–562. MR 1291244
- C. Martínez, On a stratification of the moduli $\overline {M}_{0,n}(G(2,4),d)$ and enumerative geometry, in preparation.
- M. S. Ravi, J. Rosenthal, and X. Wang, Degree of the generalized Plücker embedding of a Quot scheme and quantum cohomology, Math. Ann. 311 (1998), no. 1, 11–26. MR 1624330, DOI 10.1007/s002080050173
- Robert Speiser, Transversality theorems for families of maps, Algebraic geometry (Sundance, UT, 1986) Lecture Notes in Math., vol. 1311, Springer, Berlin, 1988, pp. 235–252. MR 951649, DOI 10.1007/BFb0082917
- I. R. Shafarevich, Basic algebraic geometry, Springer Study Edition, Springer-Verlag, Berlin-New York, 1977. Translated from the Russian by K. A. Hirsch; Revised printing of Grundlehren der mathematischen Wissenschaften, Vol. 213, 1974. MR 0447223
- Stein Arild Strømme, On parametrized rational curves in Grassmann varieties, Space curves (Rocca di Papa, 1985) Lecture Notes in Math., vol. 1266, Springer, Berlin, 1987, pp. 251–272. MR 908717, DOI 10.1007/BFb0078187
Additional Information
- Cristina Martínez
- Affiliation: Departamento de Matematicas, Facultad de Ciencias, Universidad Autonoma de Madrid, Madrid, 28049, Spain
- Email: cristina.martinez@uam.es
- Received by editor(s): June 23, 2004
- Published electronically: September 22, 2005
- © Copyright 2005 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 358 (2006), 2611-2624
- MSC (2000): Primary 14N35, 14N10; Secondary 14C05, 14C15
- DOI: https://doi.org/10.1090/S0002-9947-05-03797-9
- MathSciNet review: 2204046