Rational endomorphisms of plane preserving a rational volume form

2016/12/25 by Georgy Belousov, Belousov, Georgy
Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT)

paper · pdf · doi:10.48550/arxiv.1612.08271

openalex publication_date 2016/12/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let φ be a rational map ℙ² \dashrightarrowℙ² that preserves the rational volume form (dx)/(x)\wedge(dy)/(y). Sergey Galkin conjectured that in this case φ is necessarily birational. We show that such a map preserves the element \x,y\ of the second K-group K₂(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 φ in a way suitable for computations.

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