Algebraic varieties in the Birkhoff strata of the Grassmannian Gr ~((2)): Harrison cohomology and integrable systems

摘要

The local properties of the families of algebraic subsets W _g in the Birkhoff strata _(2g) of Gr ~((2)) containing the hyperelliptic curves of genus _g are studied. It is shown that the tangent spaces T _g for W g are isomorphic to the linear spaces of 2-coboundaries. Particular subsets in W _g are described by the integrable 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.