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 | References | Similar Articles | Additional Information

Abstract: A very general surface of degree at least four in contains no curves other than intersections with surfaces. We find a formula for the degree of the locus of surfaces in 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.

**[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****[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**- [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**, https://doi.org/10.1112/S0010437X12000437 **[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****[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**, 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**- [7]
Gerd Gotzmann,
*The irreducible components of*Hilb`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**, https://doi.org/10.1090/memo/0438- [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*, A celebration of algebraic geometry, Clay Math. Proc., vol. 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(2,1⁴)*, Math. Scand.**78**(1996), no. 1, 63–83. MR**1400851**, https://doi.org/10.7146/math.scand.a-12574 **[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**- [13]
Israel Vainsencher, computer algebra scripts,
`http://www.mat.ufmg.br/ 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.