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Quantum cohomology of rational surfaces. (English) Zbl 0843.14014
Dijkgraaf, R. H. (ed.) et al., The moduli space of curves. Proceedings of the conference held on Texel Island, Netherlands during the last week of April 1994. Basel: Birkhäuser. Prog. Math. 129, 33-80 (1995).
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 M. Kontsevich and Yu. Manin [Commun. Math. Phys. 164, No. 3, 525–562 (1994; Zbl 0853.14020)] 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.
For the entire collection see [Zbl 0827.00037].

##### MSC:
 14J26 Rational and ruled surfaces 81T70 Quantization in field theory; cohomological methods 14F99 (Co)homology theory in algebraic geometry 14N10 Enumerative problems (combinatorial problems) in algebraic geometry
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