Compositions of birational endomorphisms of the affine plane

TL;DR

This paper explores the structure and properties of birational endomorphisms of the affine plane, focusing on those defined by lines as missing or contracting curves, and examines their monoid structure.

Contribution

It introduces new results in the theory of birational endomorphisms of A^2, especially for those with line-based missing or contracting curves, and analyzes their monoid structure.

Findings

Characterization of birational endomorphisms with line missing or contracting curves

Description of the monoid structure of birational endomorphisms of A^2

New theoretical insights into the behavior of these endomorphisms

Abstract

Let A^2 denote the affine plane over an algebraically closed field of arbitrary characteristic. Besides contributing several new results in the general theory of birational endomorphisms of A^2, this article describes certain classes of birational endomorphisms f defined by requiring that the missing curves or contracting curves of f are lines. The last part of the article is concerned with the monoid structure of the set of birational endomorphisms of A^2.