Algebraic curves in Birkhoff strata of Sato Grassmannian

Details Algebraic curves in Birkhoff strata of Sato Grassmannian Algebraic curves in Birkhoff strata of Sato Grassmannian

2010-05-12

arxiv.org/abs/1005.2053v2

Physics

Mathematical Physics

38 pages, no figures, improved some definitions and lemmas, typos corrected

Scientific paper

Algebro-geometric structures arising in Birkhoff strata of Sato Grassmannian are analyzed. It is shown that each Birkhoff stratum $\Sigma_S$ contains a closed subspace $W_{\hat{S}}$ which algebraically is an infinite-dimensional commutative associative algebra and geometrically it is an infinite tower of families of algebraic curves. For the big cell the subspace $W_\varnothing$ represents the tower of families of normal rational (Veronese) curves of all orders. For $W_1$ it is the family of coordinate rings for elliptic curves. For higher strata, the subspaces $W_{1,2,...,n}$ represent families of plane $(n+1,n+2)$ curves (trigonal curves at $n=2$) and space curves of genus $n$ and index$(\bar{\partial}_{W_{1,2,...,n}})=-n$. Two methods of regularization of singular curves contained in $W_{\hat{S}}$, namely, the standard blowing-up and transition to higher strata with the change of genus are discussed. Cohomological and Poisson structures associated with the subspaces $W_{1,2,...,n}$ are considered. It is shown that the tangent bundles of the subspaces $W_{1,2,...,n}$ are isomorphic to the linear spaces of $2-$coboundaries, special class of which is provided by the systems of integrable quasilinear PDEs. For the big cell it is the dKP hierarchy. It is demonstrated also that the families of ideals for algebraic varieties in $W_{1,2,...,n}$ can be viewed as the Poisson ideals. This observation establishes a connection between families of algebraic curves in $W_{\hat{S}}$ and coisotropic deformations of such curves of zero and nonzero genus described by hierarchies of hydrodynamical type systems like dKP hierarchy. Interrelation between cohomological and Poisson structures is noted.

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