$\mathfrak{G}$-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 $[{\rm Gr}^{d, E}/\mathbb{T}_\bullet]$, 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 $\mathbb{F}^{d, E_\bullet}$.