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Linear degenerations of Schubert varieties

Giulia Iezzi

TL;DR

This work builds a framework for linear degenerations of Schubert varieties via a targeted quiver-with-relations model. By focusing on the subvariety $R^{\iota}_{\bf d}$ of representations, the authors prove every element decomposes into indecomposables $U^{(h_1,\dots,h_n)}$ and introduce two equivalent orbit parametrisations, ${\bf r}$ and ${\bf s}$, under the $B$-action. They realize Schubert varieties and their Bott–Samelson resolutions as quiver Grassmannians ${\rm Gr}_{\bf e}(M)$ with carefully chosen dimension vectors, and define the $f$-linear degenerate Schubert varieties $X^f_w$ as fibres of a universal degeneration $Y$ over $R^{\iota}_{\bf d}$. The paper also establishes a partial order on degenerations via the south-west data and proposes a conjectural flat-locus criterion linking fibre dimensions to the Schubert variety dimensions, along with a strategy to prove flatness using geometric quotients and complete-intersection arguments. These results unify representation-theoretic parametrisations with geometric degenerations, providing tools to study singularities and resolutions in a quiver-theoretic setting.

Abstract

We define linear degenerations of Schubert varieties via a special class of quiver Grassmannians. To do so, we restrict our study to an appropriate subvariety in the variety of representations of the considered quiver and describe a base change action on this subvariety. We provide two explicit parametrisations for the orbits of this action, one of which encodes the partial order relations on such orbits.

Linear degenerations of Schubert varieties

TL;DR

This work builds a framework for linear degenerations of Schubert varieties via a targeted quiver-with-relations model. By focusing on the subvariety of representations, the authors prove every element decomposes into indecomposables and introduce two equivalent orbit parametrisations, and , under the -action. They realize Schubert varieties and their Bott–Samelson resolutions as quiver Grassmannians with carefully chosen dimension vectors, and define the -linear degenerate Schubert varieties as fibres of a universal degeneration over . The paper also establishes a partial order on degenerations via the south-west data and proposes a conjectural flat-locus criterion linking fibre dimensions to the Schubert variety dimensions, along with a strategy to prove flatness using geometric quotients and complete-intersection arguments. These results unify representation-theoretic parametrisations with geometric degenerations, providing tools to study singularities and resolutions in a quiver-theoretic setting.

Abstract

We define linear degenerations of Schubert varieties via a special class of quiver Grassmannians. To do so, we restrict our study to an appropriate subvariety in the variety of representations of the considered quiver and describe a base change action on this subvariety. We provide two explicit parametrisations for the orbits of this action, one of which encodes the partial order relations on such orbits.
Paper Structure (9 sections, 19 theorems, 99 equations, 2 figures)

This paper contains 9 sections, 19 theorems, 99 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background on quiver representations
  3. The variety of $Q$-representations
  4. Schubert varieties as quiver Grassmannians
  5. The variety $R^{\iota}_{\bf d}$ of $(\Gamma,I)$-representations
  6. Two parametrisations for the representations in $R^{\iota}_{\bf d}$
  7. Linear degenerations
  8. Linear degenerations of Shubert varieties
  9. On the flat locus of linear degenerations of Schubert varieties

Key Result

Theorem 1

All representations in $R^{\iota}_{\bf d}$ can be decomposed as direct sums of the indecomposable $(\Gamma,I)$-representations $U^{(h_1,\dots,h_n)}$.

Figures (2)

  • Figure 1: The diagram formed by $M^f$, $N$ and $g$
  • Figure 2: The diagram formed by $M^f$, $N$ and $g$ in the example

Theorems & Definitions (71)

  • Theorem 1: Theorem \ref{['thm:indecompU']}
  • Theorem 2: Theorem \ref{['thm:firstparam']}
  • Theorem 3: Theorem \ref{['thm:secondparam']}
  • Theorem 4: Corollary \ref{['cor:orbitdegen']}
  • Conjecture 1: Conjecture \ref{['conj:flatlocus']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • ...and 61 more