The Isomorphism Classes of the Surfaces $x_1^{a_1}+x_2^{a_2}+x_3^{a_3}+1=0$

Abstract

Let $f=x_1^{a_1}+x_2^{a_2}+x_3^{a_3}+1\in \mathbb{C}[x_1,x_2,x_3]$ and let $g=y_1^{b_1}+y_2^{b_2}+y_3^{b_3}+1\in \mathbb{C}[y_1,y_2,y_3]$ where $a_1,a_2,a_3,b_1,b_2,b_3\ge 2$. We prove that the surfaces $V(f)\subset {\mathbb{A}}^3$ and $V(g)\subset {\mathbb{A}}^3$ are isomorphic if and only if $(a_1,a_2,a_3)=(b_1,b_2,b_3)$ up to a permutation of the entries.

Comments welcome. 15 pages. Added reference