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

We study algebraic varieties and curves arising in the Birkhoff strata of the Sato Grassmannian Gr$^{(2)}$. We show that the big cell $\Sigma_0$ contains the tower of families of the normal rational curves of all odd orders. The strata $\Sigma_{2n}$, $n=1,2,3,\dots$, contain hyperelliptic curves of genus $n$ and their coordinate rings. The strata $\Sigma_{2n+1}$, $n=0,1,2,3,\dots$, contain $(2m+1,2m+3)$ plane curves for $n=2m,2m-1$ $(m\geq2)$ and also $(3,4)$ and $(3,5)$ curves in $\Sigma_3$ and $\Sigma_5$. Curves in the strata $\Sigma_{2n+1}$ have zero genus.