Quantum cohomology of rational surfaces. (English) Zbl 0843.14014

Crauder, Bruce; Miranda, Rick

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].

Reviewer: M.Teicher (Ramat Gan)

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

Keywords:

quantum product of a rational surface; Del Pezzo surfaces

Citations:

Zbl 0853.14020