A G ] 1 N ov 2 01 1 Soliton equations and the Riemann-Schottky problem

Novikov’s conjecture on the Riemann-Schottky problem: the Jacobians of smooth algebraic curves are precisely those indecomposable principally polarized abelian varieties (ppavs) whose theta-functions provide solutions to the Kadomtsev-Petviashvili (KP) equation, was the first evidence of nowadays well-established fact: connections between the algebraic geometry and the modern theory of integrable systems is beneficial for both sides. The purpose of this paper is twofold. Our first goal is to present a proof of the strongest known characterization of a Jacobian variety in this direction: an indecomposable ppav X is the Jacobian of a curve if and only if its Kummer variety K(X) has a trisecant line [36, 37]. We call this characterization Welters’ (trisecant) conjecture after the work of Welters [64]. It was motivated by Novikov’s conjecture and

[1]  S. Grushevsky.  The Schottky Problem , 2010, 1009.0369.

[2]  I. Krichever,et al.  Abelian solutions of the soliton equations and geometry of abelian varieties. , 2008, 0804.0794.

[3]  I. Krichever,et al.  Abelian solutions of the KP equation , 2008 .

[4]  S. Grushevsky,et al.  Integrable discrete Schrodinger equations and a characterization of Prym varieties by a pair of quadrisecants , 2007, 0705.2829.

[5]  I. Krichever.  Characterizing Jacobians via trisecants of the Kummer variety , 2006 .

[6]  I. Krichever.  A characterization of Prym varieties , 2005 .

[7]  I. Krichever,et al.  Characterizing Jacobians via flexes of the Kummer variety , 2005, math/0502138.

[8]  L. Dickey.  Soliton Equations and Hamiltonian Systems , 2003 .

[9]  G. Marini.  A Geometrical Proof of Shiota's Theorem on a Conjecture of S. P. Novikov , 1998, Compositio Mathematica.

[10]  I. Krichever,et al.  Elliptic Solutions to Difference Non-Linear Equations and Related Many-Body Problems , 1998 .

[11]  I. Krichever,et al.  Symplectic forms in the theory of solitons , 1997, hep-th/9708170.

[12]  I. Krichever,et al.  Spin generalization of the Ruijsenaars-Schneider model, the non-Abelian 2D Toda chain, and representations of the Sklyanin algebra , 1995 .

[13]  R. Donagi.  Non-Jacobians in the Schottky loci , 1987 .

[14]  Takahiro Shiota,et al.  Characterization of Jacobian varieties in terms of soliton equations , 1986 .

[15]  R. Gunning.  Some Identities for Abelian Integrals , 1986 .

[16]  G. Segal,et al.  Loop groups and equations of KdV type , 1985 .

[17]  G. E. Welters.  A criterion for Jacobi varieties , 1984 .

[18]  E. Arbarello,et al.  On a Set of Equations Characterizing Riemann Matrices , 1984 .

[19]  B. Geemen,et al.  Siegel modular forms vanishing on the moduli space of curves , 1984 .

[20]  John D. Fay.  On the even-order vanishing of Jacobian theta functions , 1984 .

[21]  G. E. Welters.  On flexes of the Kummer variety (Note on a theorem of R.C. Gunning) , 1983 .

[22]  R. Gunning.  Some curves in abelian varities , 1982 .

[23]  J. Igusa.  On the irreducibility of Schottky's divisor , 1982 .

[24]  I. Krichever.  ALGEBRAIC CURVES AND NON-LINEAR DIFFERENCE EQUATIONS , 1978 .

[25]  I. V. Cherednik,et al.  Differential equations for the Baker—Akhiezer functions of algebraic curves , 1978 .

[26]  I. Krichever.  Commutative rings of ordinary linear differential operators , 1978 .

[27]  Igor Krichever,et al.  METHODS OF ALGEBRAIC GEOMETRY IN THE THEORY OF NON-LINEAR EQUATIONS , 1977 .

[28]  John D. Fay.  Theta Functions on Riemann Surfaces , 1973 .

[29]  Jean-Pierre Serre.  Faisceaux algébriques cohérents , 1955 .

[30]  H. Baker.  Note on the foregoing paper, “commutative ordinary differential operators," by J. L. Burchnall and J. W. Chaundy , 1928 .

[31]  J. L. Burchnall,et al.  Commutative Ordinary Differential Operators , 1928 .

[32]  R. Varley,et al.  The Prym Torelli problem : an update and a reformulation as a question in birational geometry , 2007 .

[33]  I. Krichever.  Integrable linear equations and the Riemann-Schottky problem , 2006 .

[34]  M. Guest.  The two-dimensional Toda lattice , 1997 .

[35]  O. Debarre.  The Schottky Problem: an Update , 1995 .

[36]  O. Debarre.  Vers une stratification de l’espace des modules des varietes abeliennes principalement polarisees , 1992 .

[37]  O. Debarre,et al.  Sur le problème de Schottky pour les variétés de Prym , 1987 .

[38]  V. Kanev.  Principal polarizations of Prym-Tjurin varieties , 1987 .

[39]  A. Beauville.  Le problème de Schottky et la conjecture de Novikov , 1987 .

[40]  Ron Donagi.  Big Schottky , 1987 .

[41]  M. Jimbo,et al.  TRANSFORMATION GROUPS FOR SOLITON EQUATIONS , 1982 .

[42]  Igor Krichever,et al.  Elliptic solutions of the Kadomtsev-Petviashvili equation and integrable systems of particles , 1980 .

[43]  Igor Krichever,et al.  Integration of nonlinear equations by the methods of algebraic geometry , 1977 .

[44]  D. Mumford,et al.  An algebro-geometric construction of commuting operators and of solutions to the Toda lattice equation , Korteweg deVries equation and related non-linear equations , 1977 .

[45]  David Mumford,et al.  Curves and their Jacobians , 1975 .

[46]  Vladimir E. Zakharov,et al.  A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I , 1974 .

[47]  D. Mumford,et al.  Prym Varieties I , 1974 .

[48]  D. Mumford.  Theta characteristics of an algebraic curve , 1971 .

[49]  D. Mumford,et al.  The irreducibility of the space of curves of given genus , 1969 .

[50]  F. Schottky.  Zur Theorie der Abelschen Functionen von vier Variabeln. , 1888 .