Prym varieties, curves with automorphisms and the Sato Grassmannian (Q1434230)

During the past fifteen years, there have been several attempts to solve a Schottky-type problem for Prym varieties, that is to characterize the locus of Prym varieties inside the corresponding moduli space of polarized abelian varieties either geometrically or by certain equations in theta constants. Some partial results concerning this problem have been obtained, in the meantime, by Shiota, Taimanov, Li, Mulase, Plaza-Martín, and others using different approaches. The paper under review also points in this direction and has two main objectives. First, the authors generalize some previous results of Shiota and Plaza-Martín to the more general case of Prym varieties associated with curves admitting an automorphism of prime order. Then they give an explicit description of the equations defining the moduli space of curves with an automorphism of prime order as a subscheme of Sato's infinite Grassmannian. Using the formal approach developed by two of the authors [\textit{J. M. Muñoz Porras} and \textit{F. J. Plaza Martín}, Equations of Hurwitz schemes in the infinite Grassmannian, Preprint http://arxiv.org/abs/math/0207091] in order to characterize Hurwitz schemes in the framework of inifinite Grassmannians, and extending it to their new concept of formal Prym varieties, the authors establish an analogue of the classical Krichever map as well as an explicit characterization of formal Prym varieties as subvarieties of the the Sato Grassmannian. Finally, in the last section of the present paper, explicit equations of the moduli spaces of curves with automorphisms of prime order are derived within the same framework. The latter formal approach is based on the results and methods of another foregoing work of two of the authors, and being concerned with the equations defining the moduli spaces of pointed curves in the infinite Grassmannian [\textit{J. M. Muñoz Porras} and \textit{F. J. Plaza Martín}, J. Differ. Geom. 51, No. 3, 431--469 (1999; Zbl 1065.14512)].