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Transactions of the American Mathematical Society

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The enumerative geometry of plane cubics. I. Smooth cubics

Author: Paolo Aluffi
Journal: Trans. Amer. Math. Soc. 317 (1990), 501-539
MSC: Primary 14N10; Secondary 14C17
MathSciNet review: 972700
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Abstract: We construct a variety of complete plane cubics by a sequence of five blow-ups over ${\mathbb {P}^9}$. This enables us to translate the problem of computing characteristic numbers for a family of plane cubics into one of computing five Segre classes, and to recover classic enumerative results of Zeuthen and Maillard.

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  • Paolo Aluffi, The characteristic numbers of smooth plane cubics, Algebraic geometry (Sundance, UT, 1986) Lecture Notes in Math., vol. 1311, Springer, Berlin, 1988, pp. 1–8. MR 951637, DOI
  • Eduardo Casas-Alvero and Sebastian Xambó-Descamps, The enumerative theory of conics after Halphen, Lecture Notes in Mathematics, vol. 1196, Springer-Verlag, Berlin, 1986. MR 852206
  • 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
  • Steven L. Kleiman and Robert Speiser, Enumerative geometry of cuspidal plane cubics, Proceedings of the 1984 Vancouver conference in algebraic geometry, CMS Conf. Proc., vol. 6, Amer. Math. Soc., Providence, RI, 1986, pp. 227–268. MR 846022
  • S. Maillard, Recherche des caractéristiques des systèmes élémentaires de courbes planes du troisième ordre, Thèses présentées á la Faculté des Sciences de Paris 39 (1871). G. Sacchiero, Numeri caratteristici delle cubiche piane cuspidali;Numeri caratteristici delle cubiche piane nodali, preprints (1985).
  • Hermann Schubert, Kalkül der abzählenden Geometrie, Springer-Verlag, Berlin-New York, 1979 (German). Reprint of the 1879 original; With an introduction by Steven L. Kleiman. MR 555576
  • U. Sterz, Berühungsvervollständigung für ebene Kurven dritter Ordnung I, Beiträge zur Algebra und Geometrie 16 (1983), 45-68, II, 17 (1984), 115-150; III, 20 (1985), 161-184; IV, 21 (1986), 91-108.
  • Israel Vainsencher, Schubert calculus for complete quadrics, Enumerative geometry and classical algebraic geometry (Nice, 1981), Progr. Math., vol. 24, Birkhäuser, Boston, Mass., 1982, pp. 199–235. MR 685770
  • S. Xambó and J. W. Miret, Fundamental numbers of cuspidal cubics; Fundamental numbers of nodal cubics, preprints (1987). H. G. Zeuthen, Détermination des caractéristiques des systèmes élémentaires de cubiques, Comptes Rendus des Séances de l’Académie des Sciences 74 (1872), 521-526, 604-607, 726-729.

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Keywords: Complete conics, complete cubics, blow-up, Segre class
Article copyright: © Copyright 1990 American Mathematical Society