## Duke Mathematical Journal

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

#### Abstract

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

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

Dates
First available in Project Euclid: 31 March 2010

https://projecteuclid.org/euclid.dmj/1270041110

Digital Object Identifier
doi:10.1215/00127094-2010-014

Mathematical Reviews number (MathSciNet)
MR2656091

Zentralblatt MATH identifier
1217.14022

#### Citation

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

#### References

• 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.