Birkhoff Strata of Sato Grassmannian and Algebraic Curves

Abstract

Algebraic and geometric structures associated with Birkhoff strata of Sato Grassmannian are analyzed. It is shown that each Birkhoff stratum ΣS contains a subset Wŝ of points for which each fiber of the corresponding tautological subbundle TBWS is closed with respect to multiplication. Algebraically TBWS is an infinite family of infinite-dimensional commutative associative algebras and geometrically it is an infinite tower of families of algebraic curves. For the big cell the subbundle TBW∅ represents the tower of families of normal rational (Veronese) curves of all degrees. For W1 such tautological subbundle is the family of coordinate rings for elliptic curves. For higher strata, the subbundles TBW1,2,..,n represent families of plane (n + 1, n + 2) curves (trigonal curves at n = 2) and space curves of genus n. Two methods of regularization of singular curves contained in TBWŝ, namely, the standard blowing-up and transition to higher strata with the change of genus are discussed.