Rational endomorphisms of plane preserving a rational volume form

TL;DR

This paper investigates rational maps of the projective plane that preserve a specific rational volume form, proving they preserve a certain K-group element and providing explicit intersection conditions for their coordinates.

Contribution

It establishes that such maps preserve a K-group element up to a constant and formulates explicit intersection conditions for computational verification.

Findings

Maps preserve the element {x,y} in K_2 up to a constant.

Explicit intersection conditions characterize these maps.

Provides a computational approach to identify such maps.

Abstract

Let $\varphi$ be a rational map $\mathbb{P}^2 \dashrightarrow\mathbb{P}^2$ that preserves the rational volume form $\frac{\mathrm{d}x}{x}\wedge\frac{\mathrm{d}y}{y}$. Sergey Galkin conjectured that in this case $\varphi$ is necessarily birational. We show that such a map preserves the element $\{x,y\}$ of the second K-group $K_2(\mathbf{k}(x,y))$ up to multiplication by a constant, and restate this condition explicitly in terms of mutual intersections of the divisors of coordinates of $\varphi$ in a way suitable for computations.