Abstract
In this article formulas for the quantum product of a rational surface are given, and used to give an algebro-geometric proof of the associativity of the quantum product for strict Del Pezzo surfaces, those for which —K is very ample. An argument for the associativity in general is proposed, which also avoids resorting to the symplectic category. The enumerative predictions of Kontsevich and Manin [3] concerning the degree of the rational curve locus in a linear system are recovered. The associativity of the quantum product for the cubic surface is shown to be essentially equivalent to the classical enumerative facts concerning lines: there are 27 of them, each meeting 10 others.
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References
A. Beauville: Compact Complex Surfaces. Cambridge University Press 1983.
A. Grassi: “Some Remarks on the Quantum Cohomology of Surfaces”, in preparation.
M. Kontsevich and Yu. Manin: “Gromov-Witten Classes, Quantum Cohomology, and Enumerative Geometry”, preprint 1994.
D. McDuff and D. Salamon: J-Holomorphic Curves and Quantum Cohomology, Univ. Lect. Series Vol. 6, American Mathematical Society, Providence 1994.
M. Reid: Undergraduate Algebraic Geometry. London Mathematical Society Student Texts Number 12, Cambridge University Press 1988.
Y. Ruan: “Topological Sigma model and Donaldson type invariants in Gromov Theory”, preprint 1993.
Y. Ruan and G. Tian: “A Mathematical Theory of Quantum Cohomology”, Math. Res. Lett. Vol. 1 (1994) 269–278.
B. Siebert and G. Tian: “On Quantum Cohomology Rings of Fano Manifolds and a Formula of Vafa and Intriligator”, preprint 1994.
E. Witten: “On the Structure of the topological phase of two-dimensional gravity”, Nuclear Physics B, Vol. 340 (1990), 281–332.
E. Witten: “Two-dimensional gravity and intersection theory on moduli space”, Surveys in Diff. Geom., Vol. 1 (1991), 243–310.
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© 1995 Birkhäuser Boston
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Crauder, B., Miranda, R. (1995). Quantum Cohomology of Rational Surfaces. In: Dijkgraaf, R.H., Faber, C.F., van der Geer, G.B.M. (eds) The Moduli Space of Curves. Progress in Mathematics, vol 129. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-4264-2_3
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DOI: https://doi.org/10.1007/978-1-4612-4264-2_3
Publisher Name: Birkhäuser Boston
Print ISBN: 978-1-4612-8714-8
Online ISBN: 978-1-4612-4264-2
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