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Enumeration of surfaces containing an elliptic quartic curve


Authors: F. Cukierman, A. F. Lopez and I. Vainsencher
Journal: Proc. Amer. Math. Soc. 142 (2014), 3305-3313
MSC (2010): Primary 14N05, 14N15; Secondary 14C05
DOI: https://doi.org/10.1090/S0002-9939-2014-11998-8
Published electronically: July 8, 2014
MathSciNet review: 3238408
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Abstract: A very general surface of degree at least four in $ \mathbb{P}^{3}$ contains no curves other than intersections with surfaces. We find a formula for the degree of the locus of surfaces in $ \mathbb{P}^{3}$ of degree at least five which contain some elliptic quartic curves. We also compute the degree of the locus of quartic surfaces containing an elliptic quartic curve, a case not covered by that formula.


References [Enhancements On Off] (What's this?)

  • [1] Allen B. Altman and Steven L. Kleiman, Foundations of the theory of Fano schemes, Compositio Math. 34 (1977), no. 1, 3-47. MR 0569043 (58 #27967)
  • [2] Israel Vainsencher and Dan Avritzer, Compactifying the space of elliptic quartic curves, Complex projective geometry (Trieste, 1989/Bergen, 1989) London Math. Soc. Lecture Note Ser., vol. 179, Cambridge Univ. Press, Cambridge, 1992, pp. 47-58. MR 1201374 (94c:14027)
  • [3] Grigoriy Blekherman, Jonathan Hauenstein, John Christian Ottem, Kristian Ranestad, and Bernd Sturmfels, Algebraic boundaries of Hilbert's SOS cones, Compos. Math. 148 (2012), no. 6, 1717-1735. MR 2999301
  • [4] Dan Edidin and William Graham, Localization in equivariant intersection theory and the Bott residue formula, Amer. J. Math. 120 (1998), no. 3, 619-636. MR 1623412 (99g:14005)
  • [5] Geir Ellingsrud and Stein Arild Strømme, Bott's formula and enumerative geometry, J. Amer. Math. Soc. 9 (1996), no. 1, 175-193. MR 1317230 (96j:14039), https://doi.org/10.1090/S0894-0347-96-00189-0
  • [6] William Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. MR 1644323 (99d:14003)
  • [7] Gerd Gotzmann, The irreducible components of Hilb $ ^{4n}(\mathbb{P}^3)$, arXiv:0811.3160v1[math.AG], 2008.
  • [8] Angelo Felice Lopez, Noether-Lefschetz theory and the Picard group of projective surfaces, Mem. Amer. Math. Soc. 89 (1991), no. 438, x+100. MR 1043786 (91f:14030)
  • [9] José Alberto Maia, Adriana Rodrigues, Fernando Xavier, and Israel Vainsencher, Enumeration of surfaces containing a curve of low degree, preprint, 2011.
  • [10] Davesh Maulik and Rahul Pandharipande, Gromov-Witten theory and Noether-Lefschetz theory (English summary). A celebration of algebraic geometry, Clay Math. Proc., 18, Amer. Math. Soc., Providence, RI, 2013, pp. 469-507. MR 3114953
  • [11] Paul Meurer, The number of rational quartics on Calabi-Yau hypersurfaces in weighted projective space $ P(2, 1^4)$, Math. Scand. 78 (1996), no. 1, 63-83. MR 1400851 (98a:14070)
  • [12] David Mumford, Lectures on curves on an algebraic surface, with a section by G. M. Bergman. Annals of Mathematics Studies, No. 59, Princeton University Press, Princeton, N.J., 1966. MR 0209285 (35 #187)
  • [13] Israel Vainsencher, computer algebra scripts, http://www.mat.ufmg.br/ $ _{\widetilde {~~}}$israel/Projetos/
    degNL (or arXiv).

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Additional Information

F. Cukierman
Affiliation: Universidad de Buenos Aires, Ciudad Universitaria, Pabellón 1, (1428) Buenos Aires, Argentina
Email: fcukier@dm.uba.ar

A. F. Lopez
Affiliation: Dipartimento di Matematica e Fisica, Università di Roma Tre, Largo San Leonardo Murialdo 1, 00146 Roma, Italy
Email: lopez@mat.uniroma3.it

I. Vainsencher
Affiliation: ICEX-Departamento de Matemática-UFMG, Av. Antônio Carlos, 6627 – Caixa Postal 702, CEP 31270-901 Belo Horizonte, MG, Brazil
Email: israel@mat.ufmg.br

DOI: https://doi.org/10.1090/S0002-9939-2014-11998-8
Keywords: Intersection theory, Noether-Lefschetz locus, enumerative geometry
Received by editor(s): November 15, 2011
Received by editor(s) in revised form: August 20, 2012, and September 19, 2012
Published electronically: July 8, 2014
Additional Notes: The first author was partially supported by CONICET-Argentina.
The second author was partially supported by PRIN Geometria delle varietà algebriche e dei loro spazi di moduli.
The third author was partially supported by CNPQ-Brasil.
Dedicated: Dedicated to Steve Kleiman on the occasion of his 70th birthday
Communicated by: Lev Borisov
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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