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

Yi Hu

TL;DR

The paper constructs a G-quotient framework for Grassmannians by a poly-diagonal torus, yielding a birational model ${f H}^{d,E_ullet}$ for ${ m Gr}^{d,E}/{oldsymbol T}_ullet$ and a G-family ${f F}^{d,E_ullet}$ parametrizing general ${oldsymbol T}_ullet$-orbit closures and their degenerations. Central to the approach is embedding open strata into products of projective spaces via explicit sets of indices, with the degenerate loci captured by multi-homogeneous binomial relations and linearized Plücker relations, leading to a blowup description of the main components. The maximal-torus case is analyzed in depth, yielding concrete equations for ${f H}^{d,E}$ and ${f F}^{d,E}_*$, as well as a stratification of these quotients by multi-linear and matroid-theoretic data. The work connects Gröbner-basis degeneration theory with Mnëv universality and offers explicit, computable equations for torus quotients of Grassmannians, with potential applications to moduli spaces and canonical birational models. The results provide a robust framework to study orbit closures, degenerations, and toric degenerations in a broad, highly structured setting.

Abstract

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 $\mathbb{F}^{d, E_\bullet}$.

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

TL;DR

The paper constructs a G-quotient framework for Grassmannians by a poly-diagonal torus, yielding a birational model

for

and a G-family

parametrizing general

-orbit closures and their degenerations. Central to the approach is embedding open strata into products of projective spaces via explicit sets of indices, with the degenerate loci captured by multi-homogeneous binomial relations and linearized Plücker relations, leading to a blowup description of the main components. The maximal-torus case is analyzed in depth, yielding concrete equations for

and

, as well as a stratification of these quotients by multi-linear and matroid-theoretic data. The work connects Gröbner-basis degeneration theory with Mnëv universality and offers explicit, computable equations for torus quotients of Grassmannians, with potential applications to moduli spaces and canonical birational models. The results provide a robust framework to study orbit closures, degenerations, and toric degenerations in a broad, highly structured setting.

Abstract

Laurent Lafforgue's presentation of a Grassmannian Gr

naturally comes equipped with the induced action of a subtorus

of PGL

. By investigating the defining ideals of

-orbit closures through general points of Gr

and studying their degenerations, we obtain a morphsim

such that

, termed the

-quotient of Gr

by

, is birational to

, and

, termed

-family of Gr

by

, is a family of general

-orbit closures and their degenerations. We obtain a series of new results on

and

.