Skip to main content
SpringerLink
Log in

Generic vanishing in characteristic \(p>0\) and the geometry of theta divisors

  • Published:
Bollettino dell'Unione Matematica Italiana Aims and scope Submit manuscript

Abstract

In this paper, we prove a strengthening of the generic vanishing result in characteristic \(p>0\) given in Hacon and Patakfalvi (Am J Math 138(4):963–998, 2016). As a consequence of this result, we show that irreducible \(\Theta \) divisors are strongly F-regular and we prove a related result for pluri-theta divisors.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Blickle, M., Schwede, K.: \(p^{-1}\)-Linear Maps in Algebra and Geometry. Commutative Algebra, pp. 123–205. Springer, New York (2013)

    MATH  Google Scholar 

  2. Ein, L., Lazarsfeld, R.: Singularities of theta divisors and the birational geometry of irregular varieties. J. Am. Math. Soc. 10(1), 243–258 (1997)

    Article  MathSciNet  Google Scholar 

  3. Green, M., Lazarsfeld, R.: Deformation theory, generic vanihsing theorems, and some conjectures of Enriques, Catanese and Beauville. Invent. Math. 90, 389–407 (1987)

    Article  MathSciNet  Google Scholar 

  4. Green, M., Lazarsfeld, R.: Higher obstructions to deforming cohomology groups of line bundles. J. Am. Math. Soc. 4, 1 (1991)

    Article  MathSciNet  Google Scholar 

  5. Grothendieck, A.: Éléments de géométrie algébrique. IV., : Étude locale des schémas et des morphismes de schémas. II. Inst. Hautes Études Sci. Publ. Math. 24, 231 (1965)

  6. Hacon, C.: Divisors on principally polarized abelian varieties. Compos. Math. 119(3), 327–335 (1999)

    Article  MathSciNet  Google Scholar 

  7. Hacon, C.: A derived category approach to generic vanishing. J. Reine Angew. Math. 575, 173–187 (2004)

    MathSciNet  MATH  Google Scholar 

  8. Hacon, C.D.: Singularities of pluri-theta divisors in Char \(p>0\). Adv. Stud. Pure Math. 65, 20 (2011)

    Google Scholar 

  9. Hacon, C., Kovács, S.: Generic vanishing fails for singular varieties in characteristic \(p\%3e0\). ; to appeared in: Recent advances in algebraic geometry: a volume in honor of Rob Lazarsfeld’s 60th birthday. Cambridge University Press. arXiv:1212.5105 (2015)

  10. Hacon, C.D., Patakfalvi, Z.: Generic vanishing in characteristic \(p>0\) and the characterization of ordinary abelian varieties. Am. J. Math. 138(4), 963–998 (2016)

    Article  MathSciNet  Google Scholar 

  11. Hacon, C.D., Patakfalvi, Z., Zhang, Lei: Birational characterization of abelian varieties and ordinary abelian varieties in characteristic \(p>0\). Duke Math. J. 168(9), 1723–1736 (2019)

    Article  MathSciNet  Google Scholar 

  12. Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, p. 52. Springer, Berlin (1977)

    Book  Google Scholar 

  13. Kollár, J.: Higher direct images of dualizing sheaves I. Ann. Math. 123, 11–42 (1986)

    Article  MathSciNet  Google Scholar 

  14. Kollár, J.: Higher direct images of dualizing sheaves II. Ann. Math. 124, 171–202 (1986)

    Article  MathSciNet  Google Scholar 

  15. Mukai, S.: Duality between \(D(X)\) and \(D({\hat{X}})\), with application to Picard sheaves. Nagoya Math. J. 81, 153–175 (1981)

    Article  MathSciNet  Google Scholar 

  16. Milicic, D.: Lectures on Derived Categories. http://www.math.utah.edu/~milicic/Eprints/dercat

  17. Pareschi, G., Popa, M.: GV-sheaves, Fourier–Mukai transform, and Generic Vanishing. Am. J. Math. 133(1), 235–271 (2011)

    Article  MathSciNet  Google Scholar 

  18. Pink, R., Roessler, D.: On \(\psi \)- invariant subvarieties of semiabelian varieties and the Manin–Mumford conjecture. J. Algebr. Geom. 13(4), 771–798 (2004)

    Article  MathSciNet  Google Scholar 

  19. Stacks Project Authors, T.: Stacks Project

  20. Schwede, K., Tucker, K.: A Survey of Test Ideals, Progress in Commutative Algebra, vol. 2, pp. 39–99. Walter de Gruyter, Berlin (2012)

    MATH  Google Scholar 

  21. Watson, A. M.: Irreducible theta divisors of principally polarized abelian varieties are strongly F-regular. arXiv:1605.09657v1

Download references

Acknowledgements

The first author was partially supported by NSF research Grants no: DMS-1952522, DMS-1801851 and by a Grant from the Simons Foundation; Award Number: 256202. The second author was partially supported by the following Grants: Grant #200021/169639 from the Swiss National Science Foundation, ERC Starting Grant #804334.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Utah, Salt Lake City, UT, 84112, USA

    Christopher D. Hacon

  2. École Polytechnique Fédérale de Lausanne (EPFL), MA C3 635, Station 8, 1015, Lausanne, Switzerland

    Zsolt Patakfalvi

Authors
  1. Christopher D. Hacon
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zsolt Patakfalvi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christopher D. Hacon.

Ethics declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Additional information

To Fabrizio Catanese on the occasion of his 70-th birthday.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hacon, C.D., Patakfalvi, Z. Generic vanishing in characteristic \(p>0\) and the geometry of theta divisors. Boll Unione Mat Ital 15, 215–244 (2022). https://doi.org/10.1007/s40574-021-00304-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40574-021-00304-6

Search

Navigation