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

Abstract

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.

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

References in corpus (5)

• Vector bundles and Lax equations on algebraic curves
• Laplacian Growth and Whitham Equations of Soliton Theory
• Coisotropic deformations of associative algebras and dispersionless integrable hierarchies
• Singular sector of the Burgers-Hopf hierarchy and deformations of hyperelliptic curves
• Birkhoff strata of the Grassmannian Gr${}^{(2)}$: Algebraic curves

Cited by in corpus (2)

• Birkhoff strata of the Grassmannian Gr${}^{(2)}$: Algebraic curves
• Cohomological, Poisson structures and integrable hierarchies in tautological subbundles for Birkhoff strata of Sato Grassmannian