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Bott's formula and enumerative geometry


Authors: Geir Ellingsrud and Stein Arild Strømme
Journal: J. Amer. Math. Soc. 9 (1996), 175-193
MSC (1991): Primary 14N10, 14C17, 14Q99; Secondary 14C05, 14L30, 14M10
MathSciNet review: 1317230
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Abstract: We outline a strategy for computing intersection numbers on
smooth varieties with torus actions using a residue formula of Bott. As an example, Gromov-Witten numbers of twisted cubic and elliptic quartic curves on some general complete intersection in projective space are computed. The results are consistent with predictions made from mirror symmetry computations. We also compute degrees of some loci in the linear system of plane curves of degrees less than 10, like those corresponding to sums of powers of linear forms, and curves carrying inscribed polygons.


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Additional Information

Geir Ellingsrud
Affiliation: Mathematical Institute University of Oslo P. O. Box 1053 N–0316 Oslo, Norway
Email: ellingsr@math.uio.no

Stein Arild Strømme
Affiliation: Mathematical Institute University of Bergen Allég 55 N–5007 Bergen, Norway
Email: stromme@mi.uib.no

DOI: http://dx.doi.org/10.1090/S0894-0347-96-00189-0
Keywords: Bott's residue formula, Gromov-Witten number, complete intersection, twisted cubic, torus action, ternary power sum, Darboux curve
Received by editor(s): August 22, 1994
Received by editor(s) in revised form: November 13, 1994
Dedicated: Dedicated to the memory of Alf Bjørn Aure, 1955–1994
Article copyright: © Copyright 1996 American Mathematical Society