Classification at infinity of polynomials of degree 3 in 3 variables

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 $homogeneouspolynomialofdegree$ 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.