Cubic threefolds and abelian varieties of dimension five
Authors:
Sebastian Casalaina-Martin and Robert Friedman
Journal:
J. Algebraic Geom. 14 (2005), 295-326
DOI:
https://doi.org/10.1090/S1056-3911-04-00379-0
Published electronically:
August 11, 2004
MathSciNet review:
2123232
Full-text PDF
Abstract |
References |
Additional Information
Abstract: This paper proves the following converse to a theorem of Mumford: Let $A$ be a principally polarized abelian variety of dimension five, whose theta divisor has a unique singular point, and suppose that the multiplicity of the singular point is three. Then $A$ is isomorphic as a principally polarized abelian variety to the intermediate Jacobian of a smooth cubic threefold. The method of proof is to analyze the possible singularities of the theta divisor of $A$, and ultimately to show that $A$ is the Prym variety of a possibly singular plane quintic.
- Arnaud Beauville, Prym varieties and the Schottky problem, Invent. Math. 41 (1977), no. 2, 149–196. MR 572974, DOI https://doi.org/10.1007/BF01418373
- Arnaud Beauville, Variétés de Prym et jacobiennes intermédiaires, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 3, 309–391 (French). MR 472843
- Arnaud Beauville, Les singularités du diviseur $\Theta $ de la jacobienne intermédiaire de l’hypersurface cubique dans ${\bf P}^{4}$, Algebraic threefolds (Varenna, 1981) Lecture Notes in Math., vol. 947, Springer, Berlin-New York, 1982, pp. 190–208 (French). MR 672617
- Arnaud Beauville, Determinantal hypersurfaces, Michigan Math. J. 48 (2000), 39–64. Dedicated to William Fulton on the occasion of his 60th birthday. MR 1786479, DOI https://doi.org/10.1307/mmj/1030132707
- C. Herbert Clemens and Phillip A. Griffiths, The intermediate Jacobian of the cubic threefold, Ann. of Math. (2) 95 (1972), 281–356. MR 302652, DOI https://doi.org/10.2307/1970801
- Ron Donagi and Roy Smith, The degree of the Prym map onto the moduli space of five-dimensional abelian varieties, Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md., 1980, pp. 143–155. MR 605340
- 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
- David Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, No. 5, Published for the Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1970. MR 0282985
- David Mumford, Prym varieties. I, Contributions to analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York, 1974, pp. 325–350. MR 0379510
Shok V. V. Shokurov, Prym varieties: theory and applications, Math USSR Izv. 23 (1984), 83–147.
- Roy Smith and Robert Varley, A Riemann singularities theorem for Prym theta divisors, with applications, Pacific J. Math. 201 (2001), no. 2, 479–509. MR 1875904, DOI https://doi.org/10.2140/pjm.2001.201.479
- Roy Smith and Robert Varley, A necessary and sufficient condition for Riemann’s singularity theorem to hold on a Prym theta divisor, Compos. Math. 140 (2004), no. 2, 447–458. MR 2027198, DOI https://doi.org/10.1112/S0010437X03000320
Tjurin A. Tjurin, The geometry of the Fano surface of a nonsingular cubic $F\subset P^4$ and Torelli theorems for Fano surfaces and cubics, Math USSR Izvestija 5 (1971), 517–546.
- Robert Varley, Weddle’s surfaces, Humbert’s curves, and a certain $4$-dimensional abelian variety, Amer. J. Math. 108 (1986), no. 4, 931–951. MR 853219, DOI https://doi.org/10.2307/2374519
Beau1 A. Beauville, Prym varieties and the Schottky problem, Invent. Math. 41 (1977), 149–196.
Beau2 A. Beauville, Variétés de Prym et Jacobiennes intermédiaires, Ann. Scient. Éc. Norm. Sup. 10 (1977), 309–391.
Beau2pt5 A. Beauville, Les singularités du diviseur $\Theta$ de la jacobienne intermédiaire de l’hypersurface cubique dans $\mathbf {P}^4$, in Algebraic Threefolds (Varenna, 1981), A. Conte (ed.), Lecture Notes in Math. 947, Springer, Berlin-New York, 1982, 190–208.
Beau3 A. Beauville, Determinantal hypersurfaces, Michigan Math. J. 48 (2000), 39–64.
CG C. H. Clemens and P. Griffiths, The intermediate Jacobian of the cubic threefold, Annals of Math. 95 (1972), 281–356.
DS R. Donagi and R. Smith, The structure of the Prym map, Acta Math. 146 (1981), 26–102.
Fulton W. Fulton, Intersection Theory, Berlin, Springer-Verlag, 1984.
Mumf1 D. Mumford, Abelian varieties, Bombay, Oxford University Press, 1970.
Mumf2 D. Mumford, Prym varieties I, in Contributions to Analysis (a collection of papers dedicated to Lipman Bers), 325–350, New York, Academic Press, 1974.
Shok V. V. Shokurov, Prym varieties: theory and applications, Math USSR Izv. 23 (1984), 83–147.
SV R. Smith and R. Varley, A Riemann singularities theorem for Prym theta divisors, with applications, Pacific J. Math. 201 (2001), 479–509.
SV2 R. Smith and R. Varley, A necessary and sufficient condition for Riemann’s singularity theorem to hold on a Prym theta divisor, Compos. Math. 140 (2004), no. 2, 447–458.
Tjurin A. Tjurin, The geometry of the Fano surface of a nonsingular cubic $F\subset P^4$ and Torelli theorems for Fano surfaces and cubics, Math USSR Izvestija 5 (1971), 517–546.
Varley R. Varley, Weddle’s surfaces, Humbert’s curves, and a certain $4$-dimensional abelian variety, Amer. J. Math. 108 (1986), 931–952.
Additional Information
Sebastian Casalaina-Martin
Affiliation:
Department of Mathematics, Columbia University, New York, New York 10027
Address at time of publication:
Department of Mathematics, SUNY Stony Brook, Stony Brook, New York 11794-3651
MR Author ID:
754836
Email:
casa@math.columbia.edu, casa@math.sunysb.edu
Robert Friedman
Affiliation:
Department of Mathematics, Columbia University, New York, New York 10027
Email:
rf@math.columbia.edu
Received by editor(s):
July 28, 2003
Published electronically:
August 11, 2004
Additional Notes:
The first author was supported in part by a VIGRE fellowship from NSF Grant #DMS-98-10750. The second author was supported in part by NSF Grant #DMS-02-00810.
