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Counting rational curves on rational surfaces. (English) Zbl 0885.14027

Grimm, Uwe (ed.) et al., Perspectives on solvable models. Dedicated to Vladimir Rittenberg on the occasion of his 60th birthday. Singapore: World Scientific. 255-276 (1994).
The recent developments in constructing topological quantum field theories have displayed a deep interrelation between physical reasoning and far-reaching new results in modern algebraic geometry. In particular, explicit intersection formulas in the enumerative geometry of moduli spaces of algebraic curves have been predicted by physicists, and the concrete problem of counting rational curves in Calabi-Yau manifolds has turned out as to be crucial for describing the so-called potential function of superconformal topological sigma-models.
The paper under review provides an enlightening, mathematically rather down-to-earth description, from a physicist’s view point, of some parts of M. Kontsevich’s pioneering recent work on the enumeration of rational curves in Calabi-Yau manifolds. For the sake of simplicity, the author restricts himself to discuss some computations towards the counting of rational curves in rational surfaces, rather than in Calabi-Yau threefolds, and that by only using elementary, classical tools from enumerative geometry. This article is addressed to physicists and those interested readers, who are less familiar with the advanced complex geometry of moduli spaces, Calabi-Yau manifolds, and mirror symmetry.
As to a deeper, more general, and more complete account on the enumerative aspects discussed here, the author refers to the recent important work “Gromov-Witten classes, quantum cohomology, and enumerative geometry” by M. Kontsevich and Yu. Manin [Commun. Math. Phys. 164, No. 3, 525-562 (1994; Zbl 0853.14020)].
For the entire collection see [Zbl 0866.00063].

MSC:

14N10 Enumerative problems (combinatorial problems) in algebraic geometry
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
14H45 Special algebraic curves and curves of low genus
14J32 Calabi-Yau manifolds (algebro-geometric aspects)

Citations:

Zbl 0853.14020
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