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

The local properties of the families of algebraic subsets Wg in the Birkhoff strata 2g of Gr(2) containing the hyperelliptic curves of genus g are studied. It is shown that the tangent spaces Tg for Wg are isomorphic to the linear spaces of 2-coboundaries. Particular subsets in Wg are described by the integrable dispersionless coupled KdV systems of hydrodynamical type defining a special class of 2-cocycles and 2-coboundaries in Tg. It is demonstrated that the blows-ups of such 2-cocycles and 2-coboundaries and gradient catastrophes for associated integrable systems are interrelated. © 2011 IOP Publishing Ltd.