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Algebraic curves in Birkhoff strata of Sato Grassmannian | | B.G. Konopelchenko
; G. Ortenzi
; | | Date: |
12 May 2010 | | Abstract: | 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$(ar{partial}_{W_{1,2,...,n}})=-n$. 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. | | Source: | arXiv, 1005.2053 | | Services: | Forum | Review | PDF | Favorites | |
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