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Izv. Akad. Nauk SSSR Ser. Mat., 1983, Volume 47, Issue 4, Pages 785–855 (Mi izv1427)  

This article is cited in 27 scientific papers (total in 28 papers)

Prym varieties: theory and applications

V. V. Shokurov

Abstract: In this paper the author determines when the principally polarized Prymian $P(\widetilde C,I)$ of a Beauville pair $(\widetilde C,I)$ satisfying a certain stability type condition is isomorphic to the Jacobian of a nonsingular curve. As an application, he points out new components in the Andreotti–Mayer variety $N_{g-4}$ of principally polarized Abelian varieties of dimension $g$ whose theta-divisors have singular locus of dimension $\geqslant g-4$; he also proves a rationality criterion for conic bundles over a minimal rational surface in terms of the intermediate Jacobian. The first part of the paper contains the necessary preliminary material introducing the reader to the modern theory of Prym varieties.
Bibliography: 32 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1984, 23:1, 83–147

Bibliographic databases:

UDC: 513.6
MSC: Primary 14H10, 14K30; Secondary 14H45
Received: 04.05.1982

Citation: V. V. Shokurov, “Prym varieties: theory and applications”, Izv. Akad. Nauk SSSR Ser. Mat., 47:4 (1983), 785–855; Math. USSR-Izv., 23:1 (1984), 83–147

Citation in format AMSBIB
\by V.~V.~Shokurov
\paper Prym varieties: theory and applications
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1983
\vol 47
\issue 4
\pages 785--855
\jour Math. USSR-Izv.
\yr 1984
\vol 23
\issue 1
\pages 83--147

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. S. Yu. Èndryushka, “Nonrationality of the general Enriques variety”, Math. USSR-Sb., 51:1 (1985), 267–273  mathnet  crossref  mathscinet  zmath
    2. S. Yu. Èndryushka, “Non-rationality of the general Enriques threefold”, Russian Math. Surveys, 39:2 (1984), 151–152  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    3. A. Alzati, M. Bertolini, “On the problem of rationality for some cubic complexes”, Indagationes Mathematicae (Proceedings), 91:4 (1988), 349  crossref
    4. V. A. Iskovskikh, “On the rationality problem for conic bundles”, Math. USSR-Sb., 72:1 (1992), 105–111  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    5. V. A. Iskovskikh, “A rationality criterion for conic bundles”, Sb. Math., 187:7 (1996), 1021–1038  mathnet  crossref  crossref  mathscinet  zmath  isi
    6. I. A. Taimanov, “Secants of Abelian varieties, theta functions, and soliton equations”, Russian Math. Surveys, 52:1 (1997), 147–218  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    7. I. A. Cheltsov, “Birationally superrigid cyclic triple spaces”, Izv. Math., 68:6 (2004), 1229–1275  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    8. I. A. Cheltsov, “The method of degeneracy and the irrationality of three-dimensional varieties with a pencil of del Pezzo surfaces”, Russian Math. Surveys, 59:4 (2004), 792–793  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    9. V. V. Przyjalkowski, I. A. Cheltsov, K. A. Shramov, “Hyperelliptic and trigonal Fano threefolds”, Izv. Math., 69:2 (2005), 365–421  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    10. I. A. Cheltsov, “Birationally rigid Fano varieties”, Russian Math. Surveys, 60:5 (2005), 875–965  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    11. M. M. Grinenko, “Fibrations into del Pezzo surfaces”, Russian Math. Surveys, 61:2 (2006), 255–300  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    12. Cheltsov I., “Nonrational nodal quartic threefolds”, Pacific Journal of Mathematics, 226:1 (2006), 65–81  crossref  isi  elib
    13. Cheltsov, I, “Nonrational del Pezzo fibrations”, Advances in Geometry, 8:3 (2008), 441  crossref  isi
    14. Yu. G. Zarhin, “Absolutely Simple Prymians of Trigonal Curves”, Proc. Steklov Inst. Math., 264 (2009), 204–215  mathnet  crossref  mathscinet  isi  elib  elib
    15. F. A. Bogomolov, Vik. S. Kulikov, Yu. I. Manin, V. V. Nikulin, D. O. Orlov, A. N. Parshin, Yu. G. Prokhorov, A. V. Pukhlikov, M. Reid, I. R. Shafarevich, V. V. Shokurov, “Vasilii Alekseevich Iskovskikh (obituary)”, Russian Math. Surveys, 64:5 (2009), 939–946  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    16. Bernardara M. Bolognesi M., “Categorical Representability and Intermediate Jacobians of Fano Threefolds”, Derived Categories in Algebraic Geometry - Tokyo 2011, EMS Ser. Congr. Rep., ed. Kawamata Y., Eur. Math. Soc., 2012, 1–25  isi
    17. Marcello Bernardara, Michele Bolognesi, “Derived categories and rationality of conic bundles”, Compositio Math, 2013, 1  crossref
    18. Asher Auel, Marcello Bernardara, Michele Bolognesi, “Fibrations in complete intersections of quadrics, Clifford algebras, derived categories, and rationality problems”, Journal de Mathématiques Pures et Appliquées, 2013  crossref
    19. Hong K., “Non-Factorial Quartic Double Solids”, Adv. Geom., 14:1 (2014), 161–174  crossref  isi
    20. V. Guletskiǐ, A. Tikhomirov, “Algebraic Cycles on Quadric Sections of Cubics in ℙ4 under the Action of Symplectomorphisms”, Proceedings of the Edinburgh Mathematical Society, 2015, 1  crossref
    21. Auel A. Bernardara M., “Cycles, Derived Categories, and Rationality”, Surveys on Recent Developments in Algebraic Geometry, Proceedings of Symposia in Pure Mathematics, 95, ed. Coskun I. DeFernex T. Gibney A., Amer Mathematical Soc, 2017, 199–266  crossref  isi
    22. Ahmadinezhad H. Okada T., “Stable Rationality of Higher Dimensional Conic Bundles”, Epijournal Geom. Algebr., 2 (2018), UNSP 5  isi
    23. Yu. G. Prokhorov, “The rationality problem for conic bundles”, Russian Math. Surveys, 73:3 (2018), 375–456  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    24. Cheltsov I. Przyjalkowski V. Shramov C., “Which Quartic Double Solids Are Rational?”, J. Algebr. Geom., 28:2 (2019), 201–243  crossref  mathscinet  zmath  isi  scopus
    25. Yuri G. Prokhorov, “Rationality of Fano Threefolds with Terminal Gorenstein Singularities. I”, Proc. Steklov Inst. Math., 307 (2019), 210–231  mathnet  crossref  crossref  isi  elib
    26. Andrew Kresch, Yuri Tschinkel, “Models of Brauer–Severi surface bundles”, Mosc. Math. J., 19:3 (2019), 549–595  mathnet  crossref
    27. Tschinkel Yu., “Rationality and Specialization”, Afr. Mat., 31:1, SI (2020), 191–205  crossref  isi
    28. P. I. Borisova, O. K. Sheinman, “Hitchin Systems on Hyperelliptic Curves”, Proc. Steklov Inst. Math., 311 (2020), 22–35  mathnet  crossref  crossref  mathscinet  isi  elib
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