##
**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.”

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.”

Reviewer: Francisco José Plaza Martín (Salamanca)

### MSC:

14H40 | Jacobians, Prym varieties |

37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |

### Citations:

Zbl 1106.14017; Zbl 0281.30013; Zbl 0485.14009; Zbl 0621.35097; Zbl 0574.14027; Zbl 1132.14032; Zbl 0679.14022; Zbl 0583.14017; Zbl 0784.14016; Zbl 1215.14031
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\textit{S. Grushevsky} and \textit{I. Krichever}, Duke Math. J. 152, No. 2, 317--371 (2010; Zbl 1217.14022)

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