Classification at infinity of polynomials of degree 3 in 3 variables

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.