## Bott’s formula and enumerative geometry

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- by Geir Ellingsrud and Stein Arild Strømme PDF
- J. Amer. Math. Soc.
**9**(1996), 175-193 Request permission

## 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.## References

- Paolo Aluffi and Carel Faber,
*Linear orbits of $d$-tuples of points in $\textbf {P}^1$*, J. Reine Angew. Math.**445**(1993), 205–220. MR**1244973** - M. F. Atiyah and R. Bott,
*The moment map and equivariant cohomology*, Topology**23**(1984), no. 1, 1–28. MR**721448**, DOI 10.1016/0040-9383(84)90021-1 - G. Ellingsrud, C. Peskine, G. Sacchiero, and S. A. Strømme (eds.),
*Complex projective geometry*, London Mathematical Society Lecture Note Series, vol. 179, Cambridge University Press, Cambridge, 1992. Papers from the Conference on Projective Varieties held in Trieste, June 19–24, 1989 and the Conference on Vector Bundles and Special Projective Embeddings held in Bergen, July 3–6, 1989. MR**1201370**, DOI 10.1017/CBO9780511662652 - W. Barth,
*Moduli of vector bundles on the projective plane*, Invent. Math.**42**(1977), 63–91. MR**460330**, DOI 10.1007/BF01389784 - M. Bershadsky, S. Cecotti, H. Ooguri, and C. Vafa,
*Holomorphic anomalies in topological field theories*, Nuclear Phys. B**405**(1993), no. 2-3, 279–304. MR**1240687**, DOI 10.1016/0550-3213(93)90548-4 - Raoul Bott,
*A residue formula for holomorphic vector-fields*, J. Differential Geometry**1**(1967), 311–330. MR**232405** - Philip Candelas, Xenia C. de la Ossa, Paul S. Green, and Linda Parkes,
*A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory*, Nuclear Phys. B**359**(1991), no. 1, 21–74. MR**1115626**, DOI 10.1016/0550-3213(91)90292-6 - James B. Carrell and David I. Lieberman,
*Holomorphic vector fields and Kaehler manifolds*, Invent. Math.**21**(1973), 303–309. MR**326010**, DOI 10.1007/BF01418791 - J. B. Carrell and D. I. Lieberman,
*Vector fields and Chern numbers*, Math. Ann.**225**(1977), no. 3, 263–273. MR**435456**, DOI 10.1007/BF01425242 - B. W. Char, K. O. Geddes, G. H. Gonnet, B. L. Leong, M. B. Monagan, and S. M. Watt,
*Maple V reference manual*, Springer-Verlag, New York, Berlin, and Heidelberg, 1991. - A. Clebsch,
*Über Curven vierter Ordnung*, J. Reine Angew. Math.**59**(1861), 125–145. - G. Ellingsrud, C. Peskine, G. Sacchiero, and S. A. Strømme (eds.),
*Complex projective geometry*, London Mathematical Society Lecture Note Series, vol. 179, Cambridge University Press, Cambridge, 1992. Papers from the Conference on Projective Varieties held in Trieste, June 19–24, 1989 and the Conference on Vector Bundles and Special Projective Embeddings held in Bergen, July 3–6, 1989. MR**1201370**, DOI 10.1017/CBO9780511662652 - Geir Ellingsrud, Ragni Piene, and Stein Arild Strømme,
*On the variety of nets of quadrics defining twisted cubics*, Space curves (Rocca di Papa, 1985) Lecture Notes in Math., vol. 1266, Springer, Berlin, 1987, pp. 84–96. MR**908709**, DOI 10.1007/BFb0078179 - Geir Ellingsrud and Stein Arild Strømme,
*On the homology of the Hilbert scheme of points in the plane*, Invent. Math.**87**(1987), no. 2, 343–352. MR**870732**, DOI 10.1007/BF01389419 - Geir Ellingsrud and Stein Arild Strømme,
*Towards the Chow ring of the Hilbert scheme of $\textbf {P}^2$*, J. Reine Angew. Math.**441**(1993), 33–44. MR**1228610** - —,
*The number of twisted cubic curves on the general quintic threefold*, Math. Scand.**7**6 (1995). - John Fogarty,
*Algebraic families on an algebraic surface*, Amer. J. Math.**90**(1968), 511–521. MR**237496**, DOI 10.2307/2373541 - William Fulton,
*Intersection theory*, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR**732620**, DOI 10.1007/978-3-662-02421-8 - B. R. Greene, D. R. Morrison, and M. R. Plesser,
*Mirror manifolds in higher dimension*, Preprint CLNS–93/1253, 1993. - Robin Hartshorne,
*Algebraic geometry*, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR**0463157**, DOI 10.1007/978-1-4757-3849-0 - S. Katz,
*Rational curves on Calabi-Yau manifolds: Verifying predictions of Mirror Symmetry*, Algebraic Geometry (E. Ballico, ed.), Marcel-Dekker, New York, 1994. - S. Katz and S. A. Strømme, “
*Schubert*”,*a Maple package for intersection theory and enumerative geometry*, Software and documentation available from the authors or by anonymous ftp from ftp.math.okstate.edu or linus.mi.uib.no, 1992. - Steven L. Kleiman,
*The transversality of a general translate*, Compositio Math.**28**(1974), 287–297. MR**360616** - M. Kontsevich,
*Enumeration of rational curves via torus actions*, Available from the e-print service at hep-th@@xxx.lanl.gov, hep-th/9405035, May 1994. - M. Kontsevich and Y. Manin,
*Gromov-Witten classes, quantum cohomology, and enumerative geometry*, Comm. Math. Phys.**164**(1994), 525–562. - J. Le Potier,
*Fibrés stables sur le plan projectif et quartiques de Lüroth*, Preprint, October 1989. - A. Libgober and J. Teitelbaum,
*Lines on Calabi-Yau complete intersections, mirror symmetry, and Picard-Fuchs equations*, Internat. Math. Res. Notices**1**(1993), 29–39. MR**1201748**, DOI 10.1155/S1073792893000030 - P. Meurer,
*The number of rational quartics on Calabi-Yau hypersurfaces in weighted projective space $\mathbf {P}(2,1^4)$*, Math. Scand.**7**8 (1996). - F. Morley,
*On the Lüroth quartic curve*, Amer. J. Math.**36**(1918). - Shigeru Mukai,
*Fano $3$-folds*, Complex projective geometry (Trieste, 1989/Bergen, 1989) London Math. Soc. Lecture Note Ser., vol. 179, Cambridge Univ. Press, Cambridge, 1992, pp. 255–263. MR**1201387**, DOI 10.1017/CBO9780511662652.018 - Ragni Piene and Michael Schlessinger,
*On the Hilbert scheme compactification of the space of twisted cubics*, Amer. J. Math.**107**(1985), no. 4, 761–774. MR**796901**, DOI 10.2307/2374355 - G. Salmon,
*Higher plane curves*, Dublin, 1875. - A. Tikhomirov and A. N. Tyurin,
*Application of geometric approximation procedure to computing the Donaldson’s polynomials for $\mathbf {CP}^2$*, Schriftenreihe Sonderforsch. Geom. Anal.**12**(1994), 1–71. - A. N. Tyurin,
*The moduli spaces of vector bundles on threefolds, surfaces and curves*I, Erlangen preprint, 1990. - I. Vainsencher,
*Enumeration of $n$-fold tangent hyperplanes to a surface*, Preprint, Universidade Federal de Pernambuco, Recife, Brasil, 1994.

## 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
- Received by editor(s): August 22, 1994
- Received by editor(s) in revised form: November 13, 1994
- © Copyright 1996 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**9**(1996), 175-193 - MSC (1991): Primary 14N10, 14C17, 14Q99; Secondary 14C05, 14L30, 14M10
- DOI: https://doi.org/10.1090/S0894-0347-96-00189-0
- MathSciNet review: 1317230

Dedicated: Dedicated to the memory of Alf Bjørn Aure, 1955–1994