Pencil of irreducible rational curves and Plane Jacobian conjecture

TL;DR

This paper investigates polynomial maps in two complex variables with irreducible rational curves and finite fibers, showing invertibility under certain conditions related to the Jacobian conjecture.

Contribution

It establishes invertibility criteria for polynomial maps with rational curves, contributing to the understanding of the Plane Jacobian conjecture.

Findings

Polynomial maps with irreducible rational curves and finite fibers are invertible if (0,0) is a regular value.

Invertibility also holds if the Jacobian determinant condition is satisfied.

Results provide new insights into the structure of polynomial maps related to the Jacobian conjecture.

Abstract

We are concerned with the behavior of the polynomial maps $F=(P,Q)$ of $\mathbb{C}^2$ with finite fibres and satisfying the condition that all of the curves $aP+bQ=0$, $(a:b)\in \mathbb{P}^1$, are irreducible rational curves. The obtained result shows that such polynomial maps $F$ is invertible if $(0,0)$ is a regular value of $F$ or if the Jacobian condition holds.