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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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  • 1 P. Aluffi and C. Faber, Linear orbits of $d$-tuples of points in $\PP^1$, J. Reine Angew. Math. 445 (1993), 205--220. MR 94j:14044
  • 2 M. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), 1--28. MR 85e:58041
  • 3 D. Avritzer and I. Vainsencher, Compactifying the space of elliptic quartic curves, in Ellingsrud et al. [], pp. 47--58. MR 93g:14002
  • 4 W. Barth, Moduli of vector bundles on the projective plane, Invent. Math. 42 (1977), 63--91. MR 57:324
  • 5 M. Bershadsky, S. Cecotti, H. Ooguri, and C. Vafa, Holomorphic anomalies in topological field theories, Nuclear Phys. B 407 (1993), 667--705. MR 94j:81254
  • 6 R. Bott, A residue formula for holomorphic vector fields, J. Differential Geom. 1 (1967), 311--330. MR 38:730
  • 7 P. Candelas, P. S. Green, X. C. de la Ossa, and L. Parkes, A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nuclear Phys. B 359 (1991), 21--74. MR 93b:32029
  • 8 J. B. Carrell and D. I. Lieberman, Holomorphic vector fields and Kaehler manifolds, Invent. Math. 21 (1973), 303--309. MR 48:4356
  • 9 ------, Vector fields and Chern numbers, Math. Ann. 225 (1977), 263--273. MR 55:8416
  • 10 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.
  • 11 A. Clebsch, Über Curven vierter Ordnung, J. Reine Angew. Math. 59 (1861), 125--145.
  • 12 G. Ellingsrud, C. Peskine, G. Sacchiero, and S. A. Strømme, editors, Complex algebraic geometry, London Math. Soc. Lecture Note Series, vol. 179, Cambridge Univ. Press, London and New York, 1992. MR 93g:14002
  • 13 G. Ellingsrud, R. Piene, and S. A. Strømme, On the variety of nets of quadrics defining twisted cubic curves, Space Curves (Franco Ghione, Christian Peskine, and Eduardo Sernesi, eds.), Lecture Notes in Math., vol. 1266, Springer-Verlag, Berlin, Heidelberg, and New York, 1987. MR 88h:14034
  • 14 G. Ellingsrud and S. A. Strømme, On the homology of the Hilbert scheme of points in the plane, Invent. Math. 87 (1987), 343--352. MR 88c:14008
  • 15 ------, Towards the Chow ring of the Hilbert scheme of $\PP^2$, J. Reine Angew. Math. 441 (1993), 33--44. MR 94i:14004
  • 16 ------, The number of twisted cubic curves on the general quintic threefold, Math. Scand. 76 (1995).
  • 17 J. Fogarty, Algebraic families on an algebraic surface, Amer. J. Math. 90 (1968), 511--521. MR 38:5778
  • 18 W. Fulton, Intersection theory, Ergeb. Math. Grenzgebiete, vol. 2, Springer-Verlag, Berlin, Heidelberg, and New York, 1984. MR 85k:14004
  • 19 B. R. Greene, D. R. Morrison, and M. R. Plesser, Mirror manifolds in higher dimension, Preprint CLNS--93/1253, 1993.
  • 20 R. Hartshorne, Algebraic geometry, Graduate Texts in Math., no. 52, Springer-Verlag, Berlin, Tokyo, and New York, 1977. MR 57:3116
  • 21 S. Katz, Rational curves on Calabi-Yau manifolds: Verifying predictions of Mirror Symmetry, Algebraic Geometry (E. Ballico, ed.), Marcel-Dekker, New York, 1994. CMP 95:04
  • 22 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 or, 1992.
  • 23 S. L. Kleiman, The transversality of a general translate, Compositio Math. 28 (1974), 287--297. MR 50:13063
  • 24 M. Kontsevich, Enumeration of rational curves via torus actions, Available from the e-print service at, hep-th/9405035, May 1994.
  • 25 M. Kontsevich and Y. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994), 525--562. CMP 94:17
  • 26 J. Le Potier, Fibrés stables sur le plan projectif et quartiques de Lüroth, Preprint, October 1989.
  • 27 A. Libgober and J. Teitelbaum, Lines on Calabi Yau complete intersections, mirror symmetry, and Picard Fuchs equations, Duke Math. J. 69 (1993), 29--39. International Mathematics Research Notes 1993, No. 1. MR 93k:14013
  • 28 P. Meurer, The number of rational quartics on Calabi-Yau hypersurfaces in weighted projective space $\PP(2,1^4)$, Math. Scand. 78 (1996).
  • 29 F. Morley, On the Lüroth quartic curve, Amer. J. Math. 36 (1918).
  • 30 S. Mukai, Fano $3$-folds, in Ellingsrud et al. [], pp. 255--263. MR 94a:14042
  • 31 R. Piene and M. Schlessinger, On the Hilbert scheme compactification of the space of twisted cubics, Amer. J. Math. 107 (1985), 761--774. MR 86m:14004
  • 32 G. Salmon, Higher plane curves, Dublin, 1875.
  • 33 A. Tikhomirov and A. N. Tyurin, Application of geometric approximation procedure to computing the Donaldson's polynomials for $\CC\PP^2$, Schriftenreihe Sonderforsch. Geom. Anal. 12 (1994), 1--71.
  • 34 A. N. Tyurin, The moduli spaces of vector bundles on threefolds, surfaces and curves I, Erlangen preprint, 1990.
  • 35 I. Vainsencher, Enumeration of $n$-fold tangent hyperplanes to a surface, Preprint, Universidade Federal de Pernambuco, Recife, Brasil, 1994.

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Additional Information

Geir Ellingsrud
Affiliation: Mathematical Institute University of Oslo P. O. Box 1053 N–0316 Oslo, Norway

Stein Arild Strømme
Affiliation: Mathematical Institute University of Bergen Allég 55 N–5007 Bergen, Norway

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