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

TL;DR

This paper investigates the local geometric properties of algebraic subsets within Birkhoff strata of the Grassmannian, revealing their connection to integrable systems and Harrison cohomology, especially focusing on hyperelliptic curves and dispersionless KdV equations.

Contribution

It establishes a link between algebraic geometry of Birkhoff strata, Harrison cohomology, and integrable hydrodynamical systems, introducing new insights into their interrelations.

Findings

Tangent spaces are isomorphic to spaces of 2-coboundaries.

Certain subsets are characterized by integrable dispersionless coupled KdV systems.

Blow-ups and gradient catastrophes are interconnected with cohomological structures.

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.