$\mathfrak{G}$-Quotients of Grassmannians and Equations

Laurent Lafforgue’s presentation of a Grassmannian Gr${}^{d,E}$ naturally comes equipped with the induced action of a subtorus ${\mathbb{T}}_{\bullet }$ of PGL$(E)$. By investigating the defining ideals of ${\mathbb{T}}_{\bullet }$-orbit closures through general points of Gr${}^{d,E}$ and studying their degenerations, we obtain a morphsim $\mathfrak{q}:{\mathbb{F}}^{d,E_{\bullet }}\to {\mathbb{H}}^{d,E_{\bullet }}$ such that ${\mathbb{H}}^{d,E_{\bullet }}$, termed the $\mathfrak{G}$-quotient of Gr${}^{d,E}$ by ${\mathbb{T}}_{\bullet }$, is birational to $[{\mathrm{G}\mathrm{r}}^{d,E}\mathrm{/}{\mathbb{T}}_{\bullet }]$, and $\mathfrak{q}$, termed $\mathfrak{G}$-family of Gr${}^{d,E}$ by ${\mathbb{T}}_{\bullet }$, is a family of general ${\mathbb{T}}_{\bullet }$-orbit closures and their degenerations. We obtain a series of new results on ${\mathbb{H}}^{d,E_{\bullet }}$ and ${\mathbb{F}}^{d,E_{\bullet }}$.