Birkhoff strata of the Grassmannian Gr$\mathrm{^{(2)}}$: Algebraic curves

TL;DR

This paper explores algebraic varieties and curves within Birkhoff strata of the Sato Grassmannian, revealing the types of curves and their properties in different strata, including rational, hyperelliptic, and plane curves.

Contribution

It characterizes the algebraic curves present in various Birkhoff strata of the Sato Grassmannian, detailing their genus and coordinate rings, and links these to the stratification structure.

Findings

The big cell contains families of normal rational curves of all odd orders.

Strata 0n contain hyperelliptic curves of genus n.

Strata 0n+1 contain specific plane curves with zero genus.

Abstract

Algebraic varieties and curves arising in Birkhoff strata of the Sato Grassmannian Gr${^{(2)}}$ are studied. It is shown that the big cell $\Sigma_0$ contains the tower of families of the normal rational curves of all odd orders. Strata $\Sigma_{2n}$, $n=1,2,3,...$ contain hyperelliptic curves of genus $n$ and their coordinate rings. Strata $\Sigma_{2n+1}$, $n=0,1,2,3,...$ contain $(2m+1,2m+3)-$plane curves for $n=2m,2m-1$ $(m \geq 2)$ and $(3,4)$ and $(3,5)$ curves in $\Sigma_3$, $\Sigma_5$ respectively. Curves in the strata $\Sigma_{2n+1}$ have zero genus.