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Mori Dream Bonds and $\mathbb{C}^*$-actions

Lorenzo Barban, Eleonora A. Romano, Luis E. Solá Conde, Stefano Urbinati

TL;DR

The paper establishes a precise correspondence between Mori dream bonds on normal projective varieties and C^*-bordisms, showing that every small modification defined by an MDB can be realized geometrically by a bordism whose chamber structure matches the associated GIT quotients. It introduces pruning as a birational, C^*-equivariant method to produce bordisms from a given action, and proves that MDBs admit chamber decompositions with chamber models corresponding to geometric quotients GX(i,i+1). A key advance is the construction of a geometric realization of MDBs: starting from an MDB (L_-,L_+), one builds a graded algebra A and a Proj X^α that carries a Bordism-friendly C^*-action; conversely, bordisms yield MDBs on the sink and source with a controlled chamber structure. The cubo-cubic Cremona example concretizes the theory by exhibiting explicit quotients and prunings, linking abstract definitions to classical birational transformations. Overall, the framework provides a systematic, equivariant lens to decompose birational maps into simpler, chamber-controlled steps with potential applications to Mori dream spaces and related birational programs.

Abstract

We construct a correspondence between Mori dream regions arising from small modifications of normal projective varieties and $\mathbb{C}^*$-actions on polarized pairs which are bordisms. Moreover, we show that the Mori dream regions constructed in this way admit a chamber decomposition on which the models are the geometric quotients of the $\mathbb{C}^*$-action. In addition we construct, from a given $\mathbb{C}^*$-action on a polarized pair for which there exist at least two admissible geometric quotients, a $\mathbb{C}^*$-equivariantly birational $\mathbb{C}^*$-variety, whose induced action is a bordism, called the pruning of the variety.

Mori Dream Bonds and $\mathbb{C}^*$-actions

TL;DR

The paper establishes a precise correspondence between Mori dream bonds on normal projective varieties and C^*-bordisms, showing that every small modification defined by an MDB can be realized geometrically by a bordism whose chamber structure matches the associated GIT quotients. It introduces pruning as a birational, C^*-equivariant method to produce bordisms from a given action, and proves that MDBs admit chamber decompositions with chamber models corresponding to geometric quotients GX(i,i+1). A key advance is the construction of a geometric realization of MDBs: starting from an MDB (L_-,L_+), one builds a graded algebra A and a Proj X^α that carries a Bordism-friendly C^*-action; conversely, bordisms yield MDBs on the sink and source with a controlled chamber structure. The cubo-cubic Cremona example concretizes the theory by exhibiting explicit quotients and prunings, linking abstract definitions to classical birational transformations. Overall, the framework provides a systematic, equivariant lens to decompose birational maps into simpler, chamber-controlled steps with potential applications to Mori dream spaces and related birational programs.

Abstract

We construct a correspondence between Mori dream regions arising from small modifications of normal projective varieties and -actions on polarized pairs which are bordisms. Moreover, we show that the Mori dream regions constructed in this way admit a chamber decomposition on which the models are the geometric quotients of the -action. In addition we construct, from a given -action on a polarized pair for which there exist at least two admissible geometric quotients, a -equivariantly birational -variety, whose induced action is a bordism, called the pruning of the variety.
Paper Structure (11 sections, 22 theorems, 54 equations, 2 figures)

This paper contains 11 sections, 22 theorems, 54 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Mori dream regions and bonds
  5. Biał ynicki-Birula decomposition, criticality
  6. GIT-quotients of ${\mathbb C}^*$-actions and admissible quotients
  7. Pruning a ${\mathbb C}^*$-action
  8. From MDBs to bordisms
  9. Construction of the geometric realization
  10. From bordisms to MDBs and the induced chamber decomposition
  11. An example: factorization of the cubo-cubic standard Cremona transformation of ${\mathbb P}^3$

Key Result

Theorem 1.2

Let $\phi:Y_-\dashrightarrow Y_+$ be a small modification defined by an MDB $(L_-,L_+)$ on a normal projective variety $Y_-$. Then there exists a normal projective variety $X$ admitting a ${\mathbb C}^*$-action which is a bordism and realizes geometrically the map $\phi$.

Figures (2)

  • Figure 1: The polytopes associated respectively to ${\mathbb P}^3,W,\widetilde{W}$ and ${\mathbb P}^3$.
  • Figure :

Theorems & Definitions (69)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • ...and 59 more