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 $\Sigma_S$ contains a subset $W_{\hat{S}}$ of points for which each fiber of the corresponding tautological subbundle $TB_{W_S}$ is closed with respect to multiplication. Algebraically $TB_{W_S}$ 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 $TB_{W_\varnothing}$ represents the tower of families of normal rational (Veronese) curves of all degrees. For $W_1$ such tautological subbundle is the family of coordinate rings for elliptic curves. For higher strata, the subbundles $TB_{W_{1,2,\dots,n}}$ 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 $TB_{W_{\hat{S}}}$, namely, the standard blowing-up and transition to higher strata with the change of genus are discussed.