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Chow quotients of $\mathbb{C}^*$-actions

Gianluca Occhetta, Eleonora A. Romano, Luis E. Solá Conde, Jarosław A. Wiśniewski

TL;DR

This work studies the Chow quotient $\mathcal{C}X$ of a ${\mathbb C}^*$-action on a smooth projective variety $X$, clarifying its relationship to GIT quotients via a detailed birational construction. Under an equalized action and a codimension condition on inner fixed-point loci, the authors show that $\mathcal{C}X$ can be obtained from the GIT quotients through a sequence of normalized blowups along centers determined by pruning birational modifications $X_{i,j}$, yielding a commutative diagram that encodes a strong factorization of the birational map between sink and source. They prove that in the convex case many intermediate Chow quotients are smooth and the pruning maps are smooth blowups, while providing explicit examples where $\mathcal{C}X$ is singular. The results illuminate how Chow quotients serve as universal limits of GIT quotients and offer a concrete, computable pipeline to study birational changes in torus actions, with connections to Hilbert quotients, the Losev–Manin spaces, and complete quadrics via explicit blowup structures. These insights contribute to a birational-geometric framework for understanding VGIT and the degeneration of cycles in torus actions, with potential applications to moduli spaces and explicit compactifications of orbit families.

Abstract

Given an action of the one-dimensional torus on a projective variety, the associated Chow quotient arises as a natural parameter space of invariant $1$-cycles, which dominates the GIT quotients of the variety. In this paper we explore the relation between the Chow and the GIT quotients of a variety, showing how to construct explicitly the former upon the latter via successive blowups under suitable assumptions. We also discuss conditions for the smoothness of the Chow quotient, and present some examples in which it is singular.

Chow quotients of $\mathbb{C}^*$-actions

TL;DR

This work studies the Chow quotient of a -action on a smooth projective variety , clarifying its relationship to GIT quotients via a detailed birational construction. Under an equalized action and a codimension condition on inner fixed-point loci, the authors show that can be obtained from the GIT quotients through a sequence of normalized blowups along centers determined by pruning birational modifications , yielding a commutative diagram that encodes a strong factorization of the birational map between sink and source. They prove that in the convex case many intermediate Chow quotients are smooth and the pruning maps are smooth blowups, while providing explicit examples where is singular. The results illuminate how Chow quotients serve as universal limits of GIT quotients and offer a concrete, computable pipeline to study birational changes in torus actions, with connections to Hilbert quotients, the Losev–Manin spaces, and complete quadrics via explicit blowup structures. These insights contribute to a birational-geometric framework for understanding VGIT and the degeneration of cycles in torus actions, with potential applications to moduli spaces and explicit compactifications of orbit families.

Abstract

Given an action of the one-dimensional torus on a projective variety, the associated Chow quotient arises as a natural parameter space of invariant -cycles, which dominates the GIT quotients of the variety. In this paper we explore the relation between the Chow and the GIT quotients of a variety, showing how to construct explicitly the former upon the latter via successive blowups under suitable assumptions. We also discuss conditions for the smoothness of the Chow quotient, and present some examples in which it is singular.
Paper Structure (17 sections, 26 theorems, 71 equations, 4 figures)

This paper contains 17 sections, 26 theorems, 71 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on ${\mathbb C}^*$-actions and GIT quotients
  3. Fixed point components
  4. Linearizations and weight maps
  5. Actions on polarized pairs
  6. Geometric quotients
  7. The smooth case
  8. Equalized actions
  9. B-type actions and bordisms
  10. A digression on Grassmannians and ${\mathbb C}^*$-actions
  11. Prunings
  12. Chow quotients
  13. The universal bundle of a ${\mathbb C}^*$-action
  14. Proof of the main statement
  15. Smoothness of the Chow quotient
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $(X,L)$ be a polarized pair, with $X$ smooth, endowed with an equalized nontrivial ${\mathbb C}^*$-action such that $\mathop{\rm codim}\nolimits(B^\pm(Y),X)>1$ for every inner fixed point component $Y$. Denote by $r$ be the criticality of the action, by $X_{i,j}$, $0\leq i<j\leq r$, the correspo

Figures (4)

  • Figure 1: SBL decomposition of $\overline{{\mathop{\rm Mov}\nolimits}(X^\flat)}\cap \langle \beta^* L,Y^\flat_0,Y^\flat_r\rangle$.
  • Figure 2: A ${\mathbb C}^*$-invariant curve $C$ is a rational normal curve in its linear span ${\mathbb P}(\mathop{\mathrm{H}}\nolimits^0(C,L_{|C}))$, and this space meets each fixed point subspace ${\mathbb P}(\mathop{\mathrm{H}}\nolimits^0(X,L)_u)$ at a point.
  • Figure 3: Very ample polytopes of the varieties $Y_-,G,Y_+$.
  • Figure 4: The ${\mathbb C}^*$-action on the variety $X$.

Theorems & Definitions (75)

  • Theorem 1.1
  • Theorem 1.2
  • Example 2.1
  • Remark 2.2
  • Lemma 2.3: AM vs. FM
  • Lemma 2.4
  • Definition 3.1
  • Remark 3.2
  • Example 3.3
  • Remark 3.4
  • ...and 65 more