Classification at infinity of polynomials of degree 3 in 3 variables

TL;DR

This paper classifies singularities at infinity for degree 3 polynomials in three variables, analyzes Milnor number jumps, and explores implications for global fibrations and fiber topology in complex three-dimensional space.

Contribution

It provides a detailed classification of infinity singularities for cubic polynomials in three variables and examines their topological and fibration properties.

Findings

Classification of singularities at infinity for degree 3 polynomials

Calculation of Milnor number jumps at infinity

Insights into global fibrations and fiber topology

Abstract

We classify singularities at infinity of polynomials of degree 3 in 3 variables, $ f (x_0, x_1, x_2) = f_1 (x_0, x_1, x_2) + f_2 (x_0, x_1, x_2) + f_3 (x_0, x_1, x_2) $, $ f_i $ homogeneous polynomial of degree $ i $, $ i = 1,2,3 $. Based on this classification, we calculate the jump in the Milnor number of an isolated singularity at infinity, when we pass from the special fiber to a generic fiber. As an application of the results, we investigate the existence of global fibrations of degree 3 polynomials in $\mathbb{C}^3$ and search for information about the topology of the fibers in each equivalence class.