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Moduli spaces for $Θ$-strata and non-reductive quotients

Ludvig Modin

TL;DR

The work develops a unified framework linking Białynicki-Birula and Θ-strata via stacks of filtrations and gradings, enabling a general non-reductive GIT theory over any affine Noetherian base. Central to the approach are the stacks of filtrations $\operatorname{Filt}(\mathcal{X})$ and gradings $\operatorname{Grad}(\mathcal{X})$ and the contracting $\mathbb{G}_m$-actions on their fibers, which yield a contracting moduli-theoretic structure formalized in Theorem A. This leads to coarse and tame moduli spaces for non-split filtrations, rigidifications, and rigid centers, and culminates in a broad $\hat{U}$-theorem that provides geometric quotients for graded unipotent actions on projective schemes, with projective quotients for appropriate unstable loci. The results apply to families and arbitrary characteristics, extending NR-GIT techniques and enabling moduli constructions for unstable objects in broader moduli problems. Altogether, the paper advances non-reductive quotient theory by situating Θ-strata and filtrations within a stack-theoretic framework that yields explicit moduli spaces and quotients, with wide potential applications in moduli problems lacking a classic GIT presentation.

Abstract

We give a new proof of the $\hat{U}$-theorem of Bérczi, Doran, Hawes and Kirwan on the existence of geometric quotients for actions of graded unipotent groups in terms of stacks of filtrations and gradings introduced by Halpern-Leistner. Our proof works over any affine Noetherian base, in particular it simultaneously generalizes the previous results to arbitrary characteristic, actions in families and to general $Θ$-strata.

Moduli spaces for $Θ$-strata and non-reductive quotients

TL;DR

The work develops a unified framework linking Białynicki-Birula and Θ-strata via stacks of filtrations and gradings, enabling a general non-reductive GIT theory over any affine Noetherian base. Central to the approach are the stacks of filtrations and gradings and the contracting -actions on their fibers, which yield a contracting moduli-theoretic structure formalized in Theorem A. This leads to coarse and tame moduli spaces for non-split filtrations, rigidifications, and rigid centers, and culminates in a broad -theorem that provides geometric quotients for graded unipotent actions on projective schemes, with projective quotients for appropriate unstable loci. The results apply to families and arbitrary characteristics, extending NR-GIT techniques and enabling moduli constructions for unstable objects in broader moduli problems. Altogether, the paper advances non-reductive quotient theory by situating Θ-strata and filtrations within a stack-theoretic framework that yields explicit moduli spaces and quotients, with wide potential applications in moduli problems lacking a classic GIT presentation.

Abstract

We give a new proof of the -theorem of Bérczi, Doran, Hawes and Kirwan on the existence of geometric quotients for actions of graded unipotent groups in terms of stacks of filtrations and gradings introduced by Halpern-Leistner. Our proof works over any affine Noetherian base, in particular it simultaneously generalizes the previous results to arbitrary characteristic, actions in families and to general -strata.

Paper Structure

This paper contains 19 sections, 30 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Algebraic spaces with a multiplicative action
  4. Equivariant mapping spaces
  5. A stacky view of Białynicki-Birula strata
  6. $\Theta$-strata and graded unipotent groups
  7. Filtrations and gradings of algebraic stacks
  8. Filtrations and gradings of quotient stacks
  9. Graded unipotent groups
  10. Moduli spaces for $\Theta$-strata
  11. Contracting action on filtrations
  12. Descending the action along rigidifications
  13. The rigidified center
  14. Proof of the main theorem
  15. Non-reductive quotients
  16. ...and 4 more sections

Key Result

Theorem A

Let $\mathcal{X}$ be an algebraic stack locally of finite presentation, with separated and quasi-compact diagonal and affine stabilizers over a quasi-separated scheme $S$. Let $\mathfrak{Z}\subset \operatorname{Grad}(\mathcal{X})$ be a connected stack of gradings of $\mathcal{X}$, then Let $\mathfrak{Z}^{\operatorname{rig}}:=\mathfrak{Z}\!\!\!\fatslash \mathcal{G}^{\operatorname{can}}$ be the rig

Theorems & Definitions (47)

  • Theorem A
  • Remark 1.1
  • Theorem B
  • Definition 2.1
  • Example 2.2
  • Theorem 2.3
  • Theorem 3.1
  • Definition 3.2
  • Remark 3.3
  • Definition 3.4
  • ...and 37 more