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

Gianluca Occhetta, Luis E. Solá Conde

TL;DR

The paper studies Chow quotients $\mathcal{C}X$ of convex varieties $X$ of Picard number one under equalized ${\mathbb C}^*$-actions with bandwidth $\delta$ and criticality $r\ge2$. It proves that $\mathcal{C}X$ is smooth, describes its boundary as a simple normal crossing divisor with $r-1$ components, and analyzes its birational geometry via partial Chow quotients (prunings) that arise from the inverse limit of GIT quotients. The nef cone is generated by the line classes $\mathcal{L}_{i,i}$ and the Mori cone by primitive curves $\Gamma_i$, with explicit contraction types (divisorial, small, or fiber-type) tied to fixed-point data; the anticanonical divisor is computed in terms of action invariants, yielding Fano, nef, or non-nef examples. The framework unifies known Chow quotients from rational homogeneous settings and provides a computable toolkit (boundary divisors, primal curves, and pruning diagrams) for understanding these quotients in broader convex settings.

Abstract

In this paper we study the Chow quotient ${\mathcal C}X$ of a convex variety $X$ of Picard number one by the action of a one dimensional torus having no non-trivial finite isotropy. Examples of these actions can be found in the rational homogeneous framework. We prove that the subvariety of ${\mathcal C}X$ parametrizing reducible torus-invariant cycles is a simple normal crossing divisor, we compute the Nef and Mori cones of ${\mathcal C}X$, and its anticanonical divisor.

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

TL;DR

The paper studies Chow quotients of convex varieties of Picard number one under equalized -actions with bandwidth and criticality . It proves that is smooth, describes its boundary as a simple normal crossing divisor with components, and analyzes its birational geometry via partial Chow quotients (prunings) that arise from the inverse limit of GIT quotients. The nef cone is generated by the line classes and the Mori cone by primitive curves , with explicit contraction types (divisorial, small, or fiber-type) tied to fixed-point data; the anticanonical divisor is computed in terms of action invariants, yielding Fano, nef, or non-nef examples. The framework unifies known Chow quotients from rational homogeneous settings and provides a computable toolkit (boundary divisors, primal curves, and pruning diagrams) for understanding these quotients in broader convex settings.

Abstract

In this paper we study the Chow quotient of a convex variety of Picard number one by the action of a one dimensional torus having no non-trivial finite isotropy. Examples of these actions can be found in the rational homogeneous framework. We prove that the subvariety of parametrizing reducible torus-invariant cycles is a simple normal crossing divisor, we compute the Nef and Mori cones of , and its anticanonical divisor.

Paper Structure

This paper contains 16 sections, 24 theorems, 48 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background material
  3. ${\mathbb C}^*$-actions
  4. Białynicki-Birula decomposition
  5. Actions on polarized pairs
  6. Equalized actions
  7. Quotients
  8. Prunings
  9. Chow quotient
  10. Chow quotients of convex varieties of Picard number one
  11. Boundary divisors
  12. Smoothness of the partial Chow quotients
  13. Primal curves
  14. Intersection numbers
  15. Nef and Mori cone of the Chow quotient
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $X$ be a convex variety of Picard number one, endowed with an equalized ${\mathbb C}^*$-action of criticality $r\geq 2$. Then the Chow quotient $\mathcal{C}\!X\space$ of $X$ by the action of ${\mathbb C}^*$ is smooth, and the subscheme of $\mathcal{C}\!X\space$ parametrizing reducible elements i

Figures (2)

  • Figure 1: The $1$-dimensional families of ${\mathbb C}^*$-invariant cycles determining the curves $\Gamma_0,\Gamma_r,\Gamma_i$, ($i=1,\dots,r-1$).
  • Figure 2: The Mori cone of $\mathcal{C}\!X\space$ and the possible types of contractions ($d$=divisorial, $f$=fiber type, $s$=small) in the cases $r=2,3,4$, with no isolated extremal fixed points.

Theorems & Definitions (60)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1: AM vs. FM
  • Corollary 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.6
  • Corollary 2.7: WORS6
  • Theorem 2.8
  • Definition 2.9
  • ...and 50 more