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Bott’s formula and enumerative geometry. (English) Zbl 0856.14019

In their work on enumerative geometry the authors have often used, in an efficient and beautiful way, that the parameter spaces that describe the problems carry a natural action of algebraic tori. This gives a powerful tool for computing the intersection ring of the parameter space, and thus to solve the enumerative problem. They have also, with success, used that the solution to many well known enumerative problems can be expressed as the top Chern class of an equivariant vector bundle on the parameter space, and that the computation of such classes often can be performed with little knowledge of the full intersection ring of the underlying parameter space. The authors also made the important observation that Bott’s residue formula gives a flexible and efficient tool for performing computations of Chern classes, because it expresses the classes in terms of local contributions on the fixed point set of the torus action.
In the present, beautiful, article the authors use all the above mentioned methods, and in particular the Bott formula, on several well known enumerative problems where the fix points are isolated. They compute the number of twisted cubic curves on certain general complete intersection Calabi-Yau threefolds, and on general hypersurfaces on degree \(n+1\) in \(\mathbb{P}^n\), with \(n\leq 8\), where, in the last case, the curves satisfy certain Schubert type conditions. Moreover the authors compute the number of elliptic quartic curves on a general hypersurface of degree \(n+1\) in \(\mathbb{P}^n\) for \(4\leq n\leq 13\). They also solve some enumerative problems concerning the variety of sums of powers of linear forms in three variables, and some problems for Darboux curves. The techniques used by the authors, and notably the use of Bott’s formula, have had considerable influence on enumerative geometry. Notably M. Kontsevich [“Enumeration of rational curves via torus actions”, in: The moduli space of curves. Proc. Conf., Texel Island 1994, Prog. Math. 129, 335-368 (1995)] has used the techniques to handle rational curves of any degree, thus dealing with non-isolated fixed points.

MSC:

14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14C05 Parametrization (Chow and Hilbert schemes)
14M10 Complete intersections

Software:

Maple; schubert
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

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