Duke Mathematical Journal

Integrable discrete Schrödinger equations and a characterization of Prym varieties by a pair of quadrisecants

Samuel Grushevsky and Igor Krichever

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We prove that Prym varieties are characterized geometrically by the existence of a symmetric pair of quadrisecant planes of the associated Kummer variety. We also show that Prym varieties are characterized by certain (new) theta-functional equations. For this purpose we construct and study a difference-differential analog of the Novikov-Veselov hierarchy

Article information

Duke Math. J., Volume 152, Number 2 (2010), 317-371.

First available in Project Euclid: 31 March 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14H40: Jacobians, Prym varieties [See also 32G20]
Secondary: 37K10: Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies (KdV, KP, Toda, etc.)


Grushevsky, Samuel; Krichever, Igor. Integrable discrete Schrödinger equations and a characterization of Prym varieties by a pair of quadrisecants. Duke Math. J. 152 (2010), no. 2, 317--371. doi:10.1215/00127094-2010-014. https://projecteuclid.org/euclid.dmj/1270041110

Export citation


  • E. Arbarello and C. De Concini, On a set of equations characterizing Riemann matrices, Ann. of Math. (2) 120 (1984), 119--140.
  • E. Arbarello, I. Krichever, and G. Marini, Characterizing Jacobians via flexes of the Kummer variety, Math. Res. Lett. 13 (2006), 109--123.
  • A. Beauville and O. Debarre, Sur le problème de Schottky pour les variétés de Prym, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1987), 613--623.
  • J. L. Burchnall and T. W. Chaundy, Commutative ordinary differential operators, I, Proc. London Math. Soc. 21 (1922), 420--440.
  • —, Commutative ordinary differential operators, II, Proc. Royal Soc. London 118 (1928), 557--583.
  • O. Debarre, ``Vers une stratification de l'espace des modules des variétés abéliennes principalement polarisées'' in Complex Algebraic Varieties (Bayreuth, Germany 1990), Lecture Notes in Math. 1507, Springer, Berlin, 1992, 71--86.
  • P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75--109.
  • A. Doliwa, P. Grinevich, M. Nieszporski, and P. M. Santini, Integrable lattices and their sublattices: From the discrete Moutard (discrete Cauchy-Riemann) 4-point equation to the self-adjoint 5-point scheme, J. Math. Phys. 48 (2007), 013513.
  • B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, The Schrödinger equation in a periodic field and Riemann surfaces, Dokl. Akad. Nauk SSSR 229 (1976), 15--18.
  • J. D. Fay, Theta Functions on Riemann Surfaces, Lecture Notes in Math. 352, Springer, Berlin, 1973.
  • —, On the even-order vanishing of Jacobian theta functions, Duke Math. J. 51 (1984), 109--132.
  • R. C. Gunning, Some curves in abelian varieties, Invent. Math. 66 (1982), 377--389.
  • I. M. Krichever, Integration of nonlinear equations by the methods of algebraic geometry, Funkcional Anal. i Priložen. 11 (1977), 15--31.
  • —, Methods of algebraic geometry in the theory of nonlinear equations, Russian Math. Surveys 32 (1977), 185--213.
  • —, Algebraic curves and non-linear difference equation, Russian Math. Surveys 33 (1978), 255--256.
  • —, Two-dimensional periodic difference operators and algebraic geometry, Soviet Math. Dokl. 32 (1985), 623--627.
  • —, A characterization of Prym varieties, Int. Math. Res. Not. 2006, art. ID 81476.
  • —, ``Integrable linear equations and the Riemann-Schottky problem'' in Algebraic Geometry and Number Theory, Progr. Math. 253, Birkhäuser, Boston (2006), 497--514.
  • —, Characterizing Jacobians via trisecants of the Kummer variety, to appear in Ann. of Math. 172 (2010), preprint.
  • I. M. Krichever and S. P. Novikov, A two-dimensional Toda chain, commuting difference operators, and holomorphic vector bundles, Russian Math. Surveys 58 (2003), 473--510.
  • I. M. Krichever and D. H. Phong, On the integrable geometry of soliton equations and $N=2$ supersymmetric gauge theories, J. Differential Geom. 45 (1997), 349--389.
  • I. M. Krichever, P. Wiegmann, and A. Zabrodin, Elliptic solutions to difference non-linear equations and related many-body problems, Comm. Math. Phys. 193 (1998), 373--396.
  • D. Mumford, Theta characteristics of an algebraic curve, Ann. Sci. École Norm. Sup. (4) 4 (1971), 181--192.
  • —, ``Prym varieties, I,'' in Contributions to Analysis (A Collection of Papers Dedicated to Lipman Bers), Academic Press, New York, 1974, 325--350.
  • —, ``An algebro-geometric construction of commuting operators and of solutions to the Toda lattice equation, Korteweg-deVries equation and related nonlinear equation'' in Proceedings of the International Symposium on Algebraic Geometry (Kyoto, 1977), Kinokuniya, Tokyo, 1978, 115--153.
  • G. Segal and G. Wilson, Loop groups and equations of KdV type, Inst. Hautes Études Sci. Publ. Math. 61 (1985), 5--65.
  • J.-P. Serre, Faisceaux algébriques cohérents, Ann. of Math. (2) 61 (1955), 197--278.
  • T. Shiota, Characterization of Jacobian varieties in terms of soliton equations, Invent. Math. 83 (1986), 333--382.
  • —, ``Prym varieties and soliton equations'' in Infinite-Dimensional Lie Algebras and Groups (Luminy-Marseille, France, 1988), Adv. Ser. Math. Phys. 7, World Sci. Publ., Teaneck, N.J., 1989, 407--448.
  • I. A. TaĭManov, Prym varieties of branched coverings and nonlinear equations, Math. USSR-Sb. 70 (1991), 367--384.
  • A. P. Veselov and S. P. Novikov, Finite-gap two-dimensional potential Schrödinger operators: Explicit formulas and evolution equations, Dokl. Akad. Nauk SSSR 279 (1984), 20--24.
  • G. E. Welters, On flexes of the Kummer variety (Note on a theorem of R. C. Gunning), Nederl. Akad. Wetensch. Indag. Math. 45 (1983), 501--520.
  • —, A criterion for Jacobi varieties, Ann. of Math. (2) 120 (1984), 497--504.