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 $a_1,a_2,a_3,b_1,b_2,b_3 \geq 2$. 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.