Francisco J. Plaza Martín

PRYM VARIETIES AND THE INFINITE GRASSMANNIAN

In this paper we study Prym varieties and their moduli space using the well-known techniques of the infinite Grassmannian. There are three main results of this paper: a new definition of the BKP hierarchy over an arbitrary base field (that generalizes the classical one over ); a characterization of Prym varieties in terms of dynamical systems, and explicit equations for the moduli space of (certain) Prym varieties. For all of these problems the language of the infinte Grassmannian, in its algebro-geometric version, allows us to deal with these problems from the same point of view.

Representations of the Witt Algebra and Gl(n)-Opers

We exhibit a new link between certain representations of the Witt algebra and some Gl(n)-opers on the punctured disc. As applications, we discuss the connection with the KdV hierarchy and Virasoro constraints and how the Virasoro constraints of the so-called topological recursion fit in our approach.

Extending Representations of \(\mathfrak {sl}(2)\) to Witt and Virasoro Algebras

We study when an 𝔰𝔩(2)

Arithmetic infinite Grassmannians and the induced central extensions

\mathfrak {sl}(2)

Coverings with prescribed ramification and Virasoro groups

-representation extends to a representation of the Witt and Virasoro algebras. We give a criterion for extendability and apply it to certain classes of weight 𝔰𝔩(2)

An Explicit Reciprocity Law Associated to Some Finite Coverings of Algebraic Curves

\mathfrak {sl}(2)

Virasoro and KdV

-modules. For all simple weight 𝔰𝔩(2)

Characterizations of Jacobians of curves with automorphisms

\mathfrak {sl}(2)

GRASSMANNIAN OF k((z)): PICARD GROUP, EQUATIONS AND AUTOMORPHISMS.

-modules and those in any of the abelian Krull-Schmidt categories of weight modules whose unique simple object is a dense module, we fully characterize which ones admit extensions, and we obtain explicit expressions for all of them. We also give partial results in the same direction for the abelian categories of weight modules which have two and three simple objects.

Equations of Hurwitz schemes in the infinite Grassmannian

The construction of families of Sato Grassmannians, their determinant line bundles and the extensions induced by them are given. The base scheme is an arbitrary scheme.