A geometric approach to the two-dimensional Jacobian Conjecture

TL;DR

This paper explores the structure of rational surfaces related to the two-dimensional Jacobian Conjecture using algebraic geometry tools, aiming to identify potential counterexamples.

Contribution

It introduces a novel geometric framework for analyzing rational maps in the context of the Jacobian Conjecture, connecting minimal model theory with combinatorial surface analysis.

Findings

Structural results on the graph of curves at infinity

Identification of a graph potentially leading to a counterexample

Insights into the combinatorial structure of rational surfaces

Abstract

Any counterexample to the two-dimensional Jacobian Conjecture gives a rational map from one projective plane to another. We use some ideas of the Minimal Model Program to study the combinatorial structure of a rational surface, that is obtained by resolving this map. Several structural results are proven, revealing a rather orderly structure of the graph of the curves at infinity. We also exhibit and discuss a graph that may lead to a counterexample to the Jacobian Conjecture.