Algebraic varieties in Birkhoff strata of the Grassmannian Gr$\mathrm{^{(2)}}$: Harrison cohomology and integrable systems

Details Algebraic varieties in Birkhoff strata of the Grassmannian Gr$\mathrm{^{(2)}}$: Harrison cohomology and integrable systems Algebraic varieties in Birkhoff strata of the Grassmannian Gr$\mathrm{^{(2)}}$: Harrison cohomology and integrable systems

2011-02-03

arxiv.org/abs/1102.0700v2

Physics

Mathematical Physics

28 pages, no figures. Generally improved version, in particular the Discussion section. Added references. Corrected typos

Scientific paper

Local properties of families of algebraic subsets $W_g$ in Birkhoff strata $\Sigma_{2g}$ of Gr$^{(2)}$ containing hyperelliptic curves of genus $g$ are studied. It is shown that the tangent spaces $T_g$ for $W_g$ are isomorphic to linear spaces of 2-coboundaries. Particular subsets in $W_g$ are described by the intergrable dispersionless coupled KdV systems of hydrodynamical type defining a special class of 2-cocycles and 2-coboundaries in $T_g$. It is demonstrated that the blows-ups of such 2-cocycles and 2-coboundaries and gradient catastrophes for associated integrable systems are interrelated.

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