Quantum Cohomology of Rational Surfaces

Details Quantum Cohomology of Rational Surfaces Quantum Cohomology of Rational Surfaces

1994-10-27

arxiv.org/abs/alg-geom/9410028v1

Mathematics

Algebraic Geometry

31 pages, AMS-LaTeX Version 1.1

Scientific paper

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 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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