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Constructing geometric realizations of birational maps between Mori Dream Spaces

Lorenzo Barban, Gianluca Occhetta, Luis E. Sol á Conde

TL;DR

We address constructing geometric realizations of birational maps between Mori Dream Spaces via faithful $C^*$-actions, proving the realizations are themselves MDSs and can realize the original map as a sequence of wall-crossings. The framework specializes to toric maps, providing a combinatorial approach and SageMath tooling for moment polytopes and Mori embeddings, and showing how to realize maps as restrictions of toric realizations after embedding MDSs in toric ambient spaces. The paper develops both general theory (dream-type maps, sharp realizations, Fano criteria) and explicit toric instances (unpruning, Fano realizations, birational contractions of a Fano fourfold) to illustrate the constructions. It also explains how to extend these toric realizations to arbitrary MDSs via Mori embeddings and restricting toric geometry, accompanied by concrete examples. Overall, the work links $C^*$-equivariant MMPs with explicit geometric realizations, providing practical tools for analyzing birational maps between MDSs and their toric counterparts.

Abstract

We construct geometric realizations -- projective algebraic versions of cobordisms -- for birational maps between Mori Dream Spaces. We show that these geometric realizations are Mori Dream Spaces, as well, and that they can be constructed so that they induce factorizations of the original birational maps as compositions of wall-crossings. In the case of toric birational maps between normal $\mathbb{Q}$-factorial, projective toric varieties, we provide several SageMath functions to work with $\mathbb{C}^*$-actions and birational geometry; in particular we show how to explicitly construct a moment polytope of a toric geometric realization. Moreover, by embedding Mori Dream Spaces in toric varieties, we obtain geometric realizations of birational maps of Mori Dream Spaces as restrictions of toric geometric realizations. We also provide examples and discuss when a geometric realization is Fano.

Constructing geometric realizations of birational maps between Mori Dream Spaces

TL;DR

We address constructing geometric realizations of birational maps between Mori Dream Spaces via faithful -actions, proving the realizations are themselves MDSs and can realize the original map as a sequence of wall-crossings. The framework specializes to toric maps, providing a combinatorial approach and SageMath tooling for moment polytopes and Mori embeddings, and showing how to realize maps as restrictions of toric realizations after embedding MDSs in toric ambient spaces. The paper develops both general theory (dream-type maps, sharp realizations, Fano criteria) and explicit toric instances (unpruning, Fano realizations, birational contractions of a Fano fourfold) to illustrate the constructions. It also explains how to extend these toric realizations to arbitrary MDSs via Mori embeddings and restricting toric geometry, accompanied by concrete examples. Overall, the work links -equivariant MMPs with explicit geometric realizations, providing practical tools for analyzing birational maps between MDSs and their toric counterparts.

Abstract

We construct geometric realizations -- projective algebraic versions of cobordisms -- for birational maps between Mori Dream Spaces. We show that these geometric realizations are Mori Dream Spaces, as well, and that they can be constructed so that they induce factorizations of the original birational maps as compositions of wall-crossings. In the case of toric birational maps between normal -factorial, projective toric varieties, we provide several SageMath functions to work with -actions and birational geometry; in particular we show how to explicitly construct a moment polytope of a toric geometric realization. Moreover, by embedding Mori Dream Spaces in toric varieties, we obtain geometric realizations of birational maps of Mori Dream Spaces as restrictions of toric geometric realizations. We also provide examples and discuss when a geometric realization is Fano.

Paper Structure

This paper contains 17 sections, 9 theorems, 31 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. ${\mathbb C}^*$-action on projective varieties
  5. Geometric quotients and the associated birational map of a ${\mathbb C}^*$-action
  6. Toric varieties
  7. Geometric realizations of birational maps
  8. Geometric realizations of maps of dream type
  9. Proof of Theorem \ref{['thm:main']}
  10. When are geometric realizations Fano?
  11. Geometric realizations of toric birational maps
  12. Unpruning
  13. An example: representing birational contractions of a Fano fourfold
  14. Geometric realizations of birational maps between MDSs
  15. Mori embeddings
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $\phi:Y_-\dashrightarrow Y_+$ be the natural birational map between two ${\mathbb{Q}}$-factorial birational contractions of a Mori Dream Space $Y$. Then there exists a sharp geometric realization of $\phi$ that is a Mori Dream Space.

Figures (4)

  • Figure 1: Moment polytopes of the varieties ${\mathbb P}^3$, $\mathcal{G}\!X_2,\mathcal{G}\!X_3,{\mathbb P}^3$.
  • Figure 2: The Mori chamber decomposition of the Fano $4$-fold $Y$.
  • Figure 3: Two geometric realizations of the birational map between two birational contractions of the Fano $4$-fold $Y$.
  • Figure 4: The effective cone of ${\mathdutchcal Y}_-$ and its Mori chamber decomposition.

Theorems & Definitions (22)

  • Theorem 1.1
  • Proposition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • Definition 3.4
  • ...and 12 more