Structure of the Newton tree at infinity of a polynomial in two variables

TL;DR

This paper investigates the structure of the Newton tree at infinity for polynomial maps in two variables, analyzing its complexity through the genus of the generic fiber, within a specific compactification framework.

Contribution

It provides a detailed analysis of the Newton tree at infinity for polynomials in two variables, relating its structure to the genus of the generic fiber.

Findings

The dual graph of the divisor at infinity is a tree.

The complexity of the Newton tree correlates with the genus of the generic fiber.

The paper characterizes the structure of the Newton tree in this setting.

Abstract

Let $f:\mathbb{C}^2 \to \mathbb{C}$ be a polynomial map. Let $\mathbb{C}^2 \subset X$ be a compactification of $\mathbb{C}^2$ where $X$ is a smooth rational compact surface and such that there exists a morphism of varieties $\Phi :X\to \mathbb{P}^1$ which extends $f$. Put $\mathcal{D}=X\setminus \mathbb{C}^2$; $\mathcal{D}$ is a curve whose irreducible components are smooth rational compact curves and all its singularities are ordinary double points. The dual graph of $\mathcal{D}$ is a tree. We are interested in this tree, and we analyse its complexity in terms of the genus of the generic fiber of $f$.