Theta divisors with curve summands and the Schottky problem

Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review

Autoren

  • Stefan Schreieder

Externe Organisationen

  • Max-Planck-Institut für Mathematik
  • Rheinische Friedrich-Wilhelms-Universität Bonn
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Details

OriginalspracheEnglisch
Seiten (von - bis)1017-1039
Seitenumfang23
FachzeitschriftMathematische Annalen
Jahrgang365
Ausgabenummer3-4
PublikationsstatusVeröffentlicht - 1 Aug. 2016
Extern publiziertJa

Abstract

We prove the following converse of Riemann’s Theorem: let (A, Θ) be an indecomposable principally polarized abelian variety whose theta divisor can be written as a sum of a curve and a codimension two subvariety Θ = C+ Y. Then C is smooth, A is the Jacobian of C, and Y is a translate of Wg - 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.

ASJC Scopus Sachgebiete

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Theta divisors with curve summands and the Schottky problem. / Schreieder, Stefan.
in: Mathematische Annalen, Jahrgang 365, Nr. 3-4, 01.08.2016, S. 1017-1039.

Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review

Schreieder S. Theta divisors with curve summands and the Schottky problem. Mathematische Annalen. 2016 Aug 1;365(3-4):1017-1039. doi: 10.1007/s00208-015-1287-8
Schreieder, Stefan. / Theta divisors with curve summands and the Schottky problem. in: Mathematische Annalen. 2016 ; Jahrgang 365, Nr. 3-4. S. 1017-1039.
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abstract = "We prove the following converse of Riemann{\textquoteright}s Theorem: let (A, Θ) be an indecomposable principally polarized abelian variety whose theta divisor can be written as a sum of a curve and a codimension two subvariety Θ = C+ Y. Then C is smooth, A is the Jacobian of C, and Y is a translate of Wg - 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.",
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N1 - Funding information: 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 []. 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.

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AB - We prove the following converse of Riemann’s Theorem: let (A, Θ) be an indecomposable principally polarized abelian variety whose theta divisor can be written as a sum of a curve and a codimension two subvariety Θ = C+ Y. Then C is smooth, A is the Jacobian of C, and Y is a translate of Wg - 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.

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KW - DPC Problem

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