Abstract
We prove the following converse of Riemann’s Theorem: let \((A,\Theta )\) be an indecomposable principally polarized abelian variety whose theta divisor can be written as a sum of a curve and a codimension two subvariety \(\Theta =C+Y\). Then C is smooth, A is the Jacobian of C, and Y is a translate of \(W_{g-2}(C)\). As applications, we determine all theta divisors that are dominated by a product of curves and characterize Jacobians by the existence of a d-dimensional subvariety with curve summand whose twisted ideal sheaf is a generic vanishing sheaf.
Similar content being viewed by others
Notes
A priori \(n\ge \dim (X)\), but by [21, Lem. 6.1], we may actually assume \(n=\dim (X)\).
In fact, Pareschi and Popa treat the more general case of an equidimensional closed reduced subscheme \(Z\subseteq A\), but for our purposes the case of subvarieties will be sufficient.
References
Arbarello, E., Cornalba, M., Griffiths, P.A., Harris, J.: Geometry of Algebraic Curves I. Springer, New York (1985)
Birkenhake, C., Lange, H.: Complex Abelian Varieties, 2nd edn. Springer, New York (2004)
Clemens, C.H., Griffiths, P.A.: The intermediate Jacobian of the cubic threefold. Ann. Math. 95, 281-356 (1972)
Debarre, O.: Minimal cohomology classes and Jacobians. J. Algebraic Geom. 4(2), 321-335 (1995)
Debarre, O.: Tores et variétés abéliennes complex, Cours Spécialisés 6. Société Mathématique de France, EDP Sciences, Paris (1999)
Deligne, P.: La conjecture de Weil pour les surfaces K3. Invent. Math. 15, 206-226 (1972)
Ein, L., Lazarsfeld, R.: Singularities of theta divisors and the birational geometry of irregular varieties. J. Am. Math. Soc. 10, 243-258 (1997)
Gunning, R.C.: Some curves in abelian varieties. Invent. Math. 66, 377-389 (1982)
Grushevsky, S.: The Schottky problem. In: Current Developments in Algebraic Geometry, MSRI Publications, vol. 59, pp. 129-164. Cambridge University Press, Cambridge (2012)
Hoyt, W.L.: On products and algebraic families of Jacobian varieties. Ann. Math. 77, 415-423 (1963)
Huybrechts, D.: Fourier-Mukai Transforms in Algebraic Geometry. Oxford Mathematical Monographs, Oxford (2006)
Krichever, I.: Characterizing Jacobians via trisecants of the Kummer variety. Ann. Math. 172, 485-516 (2010)
Little, J.: Correction to: On Lie’s approach to the study of translation manifolds. http://mathcs.holycross.edu/ little/Corrs.html (on his personal webpage)
Little, J.: On Lie’s approach to the study of translation manifolds. J. Differ. Geom. 26, 253-272 (1987)
Matsusaka, T.: On a characterization of a Jacobian variety. Mem. Coll. Sci. Kyoto Ser. A Math. 32, 1-19 (1959)
Martens, H.H.: An extended Torelli theorem. Am. J. Math. 87, 257-261 (1965)
Pareschi, G., Popa, M.: Generic vanishing and minimal cohomology classes on abelian varieties. Math. Ann. 340, 209-222 (2008)
Ran, Z.: A characterization of five-dimensional Jacobian varieties. Invent. Math. 67, 395-422 (1982)
Ran, Z.: On a theorem of Martens. Rend. Sem. Mat. Univers. Politec. Torino 44, 287-291 (1986)
Ran, Z.: On subvarieties of abelian varieties. Invent. Math. 62, 459-479 (1981)
Schoen, C.: Varieties dominated by product varieties. Int. J. Math. 7, 541-571 (1996)
Serre, J.-P.: Letter to Grothendieck, March 31. In: Grothendieck-Serre Correspondence, p. 2004. AMS, Providence (1964)
Weil, A.: Zum Beweis des Torellischen Satzes. Nachr. Akad. Wiss. Göttingen, 33-53 (1957)
Welters, G.E.: A characterization of non-hyperelliptic Jacobi varieties. Invent. Math. 74, 437-440 (1983)
Acknowledgments
I would like to thank my advisor D. Huybrechts for constant support, encouragement and discussions about the DPC problem. Thanks go also to C. Schnell for his lectures on generic vanishing theory, held in Bonn during the winter semester 2013/14, where I learned about GV-sheaves and Ein–Lazarsfeld’s result [7]. I am grateful to J. Fresan, D. Kotschick, L. Lombardi and M. Popa for useful comments. Special thanks to the anonymous referee for helpful comments and corrections. The author is member of the BIGS and the SFB/TR 45 and supported by an IMPRS Scholarship of the Max Planck Society.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Schreieder, S. Theta divisors with curve summands and the Schottky problem. Math. Ann. 365, 1017–1039 (2016). https://doi.org/10.1007/s00208-015-1287-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-015-1287-8