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On a Theorem by Lin and Shinder through the Lens of Median Geometry

Anthony Genevois, Anne Lonjou, Christian Urech

TL;DR

The paper reframes Lin and Shinder's result on nontrivial homomorphisms from Cremona groups into $\mathbb{Z}$ via median geometry. It develops a general method to produce homomorphisms from groups acting on median graphs to right-angled Artin groups, and then applies this to Cremona groups by constructing a median graph $\mathcal{C}^0(X)$ with a faithful isometric action of $\operatorname{Bir}(X)$. A key outcome is a natural map $\varphi: \operatorname{Bir}(X)\to \mathbb{Z}[\operatorname{Div}(X)/_\approx]$, with kernel containing all pseudo-regularisable transformations, yielding nontrivial obstructions in several parameter regimes for $\operatorname{Bir}(\mathbb{P}^n_k)$. This provides a self-contained, geometry-driven derivation of Lin–Shinder-type results, clarifying the role of Cremona equivalence of divisors in birational dynamics.

Abstract

Recently, Lin and Shinder constructed non-trivial homomorphisms from Cremona groups of rank >2 to \mathbb{Z} using motivic techniques. In this short note we propose an alternative perspective from median geometry on their theorem.

On a Theorem by Lin and Shinder through the Lens of Median Geometry

TL;DR

The paper reframes Lin and Shinder's result on nontrivial homomorphisms from Cremona groups into via median geometry. It develops a general method to produce homomorphisms from groups acting on median graphs to right-angled Artin groups, and then applies this to Cremona groups by constructing a median graph with a faithful isometric action of . A key outcome is a natural map , with kernel containing all pseudo-regularisable transformations, yielding nontrivial obstructions in several parameter regimes for . This provides a self-contained, geometry-driven derivation of Lin–Shinder-type results, clarifying the role of Cremona equivalence of divisors in birational dynamics.

Abstract

Recently, Lin and Shinder constructed non-trivial homomorphisms from Cremona groups of rank >2 to \mathbb{Z} using motivic techniques. In this short note we propose an alternative perspective from median geometry on their theorem.
Paper Structure (6 sections, 9 theorems, 7 equations, 1 figure)

This paper contains 6 sections, 9 theorems, 7 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Acknowledgments.
  3. Median graphs
  4. Homomorphisms to right-angled Artin groups
  5. Actions of Cremona groups on Median graphs
  6. Proof of Theorem \ref{['thm:quotient']}

Key Result

Theorem 1.1

Let $X$ be a normal variety and let $f\in\operatorname{Bir}(X)$. Let $H_1,\dots, H_k$ be the irreducible components of strict codimension 1 of the exceptional locus of $f$ and let $K_1,\dots, K_m$ be the irreducible components of strict codimension 1 of the exceptional locus of $f^{-1}$. The assignm

Figures (1)

  • Figure 1: In this example, the hyperplane defined by the red edges is transverse to the hyperplanes defined by the green and yellow edges.

Theorems & Definitions (14)

  • Theorem 1.1
  • Theorem 1.2: lin2022motivic
  • Remark 1.1
  • Corollary 1.3
  • Lemma 2.1
  • proof : Proof of Lemma \ref{['lem:PathMedian']}.
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 4 more