×

Integrable discrete Schrödinger equations and a characterization of Prym varieties by a pair of quadrisecants. (English) Zbl 1217.14022

From the introduction: “The problem of characterizing the locus \({\mathcal P}_g\) of Pryms of dimension \(g\) in the moduli space \({\mathcal A}_g\) of all principally polarized abelian varieties (PPAVs) is well known and has attracted a lot of interest over the years. In some sense the Prym varieties may be geometrically the easiest to understand PPAVs beyond Jacobians, and one could hope that studying them would be a first step towards understanding the geometry of more general abelian varieties as well.
Recently, the Prym varieties for the case of an involution with two fixed points were characterized by I. Krichever [Int. Math. Res. Not. 2006, No. 11, Article ID 86295 (2006; Zbl 1106.14017)] by the property of the theta function satisfying a certain partial differential equation, coming from the theory of integrable Schrödinger equations. The main goal of this paper is to give a geometric (and equivalent theta-functional) characterization of Prym varieties corresponding to involutions with no fixed points (the Prym varieties of involutions with two fixed points arise as a degeneration of this case).
A projective \((m-2)\)-dimensional plane \({\mathbb P}_{m-2}\subset{\mathbb P}_{2^g-1}\) intersecting \(K(X)\) in at least \(m\) points is called an \(m\)-secant of the Kummer variety.
The Kummer images of Jacobians of curves were shown to admit many trisecant lines [J. D. Fay, Lect. Notes Math. 352 (1973; Zbl 0281.30013)]. It was then shown by R. C. Gunning [Invent. Math. 66, 377–389 (1982; Zbl 0485.14009)] that the existence of a one-dimensional family of trisecants in fact suffices to characterize Jacobians among all PPAVs. Welters, inspired by Gunning’s theorem and Novikov’s conjecture proved later by T. Shiota [Invent. Math. 83, 333–382 (1986; Zbl 0621.35097)], formulated in [G. E. Welters, Ann. Math. (2) 120, 497–504 (1984; Zbl 0574.14027)] the following conjecture: If \(K(X)\) has a trisecant, and \(X\) is indecomposable, then \(X\) is a Jacobian. This was recently proved by I. Krichever [Prog. Math. 253, 497–514 (2006; Zbl 1132.14032); Ann. Math. (2) 172, No. 1, 485–516 (2010; Zbl 1215.14031)].
Prym varieties possess generalizations of some properties that Jacobians possess. A. Beauville and O. Debarre [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 14, No. 4, 613–623 (1987; Zbl 0679.14022)] and J. Fay [Duke Math. J. 51, 109–132 (1984; Zbl 0583.14017)] showed that the Kummer images of Prym varieties admit many quadrisecant planes. Similarly to the Jacobian case, it was then shown by O. Debarre [Lect. Notes Math. 1507, 71–86 (1992; Zbl 0784.14016)] that the existence of a one-dimensional family of quadrisecants characterizes Pryms. However, Beauville and Debarre showed that the existence of a single quadrisecant plane to the Kummer variety does not characterize Pryms.
In this paper we prove that the Prym varieties are characterized by the existence of a symmetric pair of quadrisecants of the corresponding Kummer variety – i.e. of two different 2-planes each intersecting the Kummer variety in 4 points, such that the points of secancy for the two planes are related in some precise way. We deduce this from a characterization by some theta-functional equations, and study the associated discrete Schrödinger equations along the way.”

MSC:

14H40 Jacobians, Prym varieties
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] E. Arbarello and C. De Concini, On a set of equations characterizing Riemann matrices , Ann. of Math. (2) 120 (1984), 119–140. JSTOR: · Zbl 0551.14016 · doi:10.2307/2007073
[2] E. Arbarello, I. Krichever, and G. Marini, Characterizing Jacobians via flexes of the Kummer variety , Math. Res. Lett. 13 (2006), 109–123. · Zbl 1098.14020 · doi:10.4310/MRL.2006.v13.n1.a9
[3] 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. · Zbl 0679.14022
[4] J. L. Burchnall and T. W. Chaundy, Commutative ordinary differential operators, I , Proc. London Math. Soc. 21 (1922), 420–440. · JFM 49.0311.03
[5] -, Commutative ordinary differential operators, II , Proc. Royal Soc. London 118 (1928), 557–583. · JFM 54.0439.01
[6] 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. · Zbl 0784.14016 · doi:10.1007/BFb0094511
[7] 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. · Zbl 0181.48803 · doi:10.1007/BF02684599
[8] 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. · Zbl 1121.37058 · doi:10.1063/1.2406056
[9] 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. · Zbl 0441.35021
[10] J. D. Fay, Theta Functions on Riemann Surfaces , Lecture Notes in Math. 352 , Springer, Berlin, 1973. · Zbl 0281.30013 · doi:10.1007/BFb0060090
[11] -, On the even-order vanishing of Jacobian theta functions , Duke Math. J. 51 (1984), 109–132. · Zbl 0583.14017 · doi:10.1215/S0012-7094-84-05106-8
[12] R. C. Gunning, Some curves in abelian varieties, Invent. Math. 66 (1982), 377–389. · Zbl 0485.14009 · doi:10.1007/BF01389218
[13] I. M. Krichever, Integration of nonlinear equations by the methods of algebraic geometry , Funkcional Anal. i Priložen. 11 (1977), 15–31.
[14] -, Methods of algebraic geometry in the theory of nonlinear equations , Russian Math. Surveys 32 (1977), 185–213. · Zbl 0386.35002 · doi:10.1070/RM1977v032n06ABEH003862
[15] -, Algebraic curves and non-linear difference equation , Russian Math. Surveys 33 (1978), 255–256. · Zbl 0412.39002 · doi:10.1070/RM1978v033n04ABEH002503
[16] -, Two-dimensional periodic difference operators and algebraic geometry , Soviet Math. Dokl. 32 (1985), 623–627. · Zbl 0603.39004
[17] -, A characterization of Prym varieties , Int. Math. Res. Not. 2006 , art. ID 81476. · Zbl 1106.14017 · doi:10.1155/IMRN/2006/81476
[18] -, “Integrable linear equations and the Riemann-Schottky problem” in Algebraic Geometry and Number Theory , Progr. Math. 253 , Birkhäuser, Boston (2006), 497–514. · Zbl 1132.14032 · doi:10.1007/978-0-8176-4532-8_8
[19] -, Characterizing Jacobians via trisecants of the Kummer variety , to appear in Ann. of Math. 172 (2010), · Zbl 1215.14031 · doi:10.4007/annals.2010.172.485
[20] 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. · Zbl 1060.37068 · doi:10.1070/RM2003v058n03ABEH000628
[21] 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. · Zbl 0889.58044
[22] 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. · Zbl 0907.35124 · doi:10.1007/s002200050333
[23] D. Mumford, Theta characteristics of an algebraic curve , Ann. Sci. École Norm. Sup. (4) 4 (1971), 181–192. · Zbl 0216.05904
[24] -, “Prym varieties, I,” in Contributions to Analysis (A Collection of Papers Dedicated to Lipman Bers) , Academic Press, New York, 1974, 325–350. · Zbl 0299.14018
[25] -, “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. · Zbl 0423.14007
[26] G. Segal and G. Wilson, Loop groups and equations of KdV type , Inst. Hautes Études Sci. Publ. Math. 61 (1985), 5–65. · Zbl 0592.35112 · doi:10.1007/BF02698802
[27] J.-P. Serre, Faisceaux algébriques cohérents , Ann. of Math. (2) 61 (1955), 197–278. JSTOR: · Zbl 0067.16201 · doi:10.2307/1969915
[28] T. Shiota, Characterization of Jacobian varieties in terms of soliton equations , Invent. Math. 83 (1986), 333–382. · Zbl 0621.35097 · doi:10.1007/BF01388967
[29] -, “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. · Zbl 0766.14020
[30] I. A. TaĭManov, Prym varieties of branched coverings and nonlinear equations , Math. USSR-Sb. 70 (1991), 367–384. · Zbl 0732.35082 · doi:10.1070/SM1991v070n02ABEH001257
[31] 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. · Zbl 0613.35020
[32] 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. · Zbl 0542.14029
[33] -, A criterion for Jacobi varieties , Ann. of Math. (2) 120 (1984), 497–504. JSTOR: · Zbl 0574.14027 · doi:10.2307/1971084
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.