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

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.

💡 Research Summary

The paper investigates algebraic varieties and curves that naturally arise in the Birkhoff strata of the Sato Grassmannian Gr^{(2)}. The authors begin by recalling that Gr^{(2)} consists of two‑dimensional subspaces W of the infinite‑dimensional vector space V=ℂ((z)), equipped with the standard splitting V=V_{+}⊕V_{-}, where V_{+}=ℂ