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

TL;DR

This paper classifies when surfaces defined by sums of three monomials plus one are isomorphic, showing they are isomorphic if and only if their exponents match up to permutation.

Contribution

It provides a complete characterization of isomorphism classes for a family of algebraic surfaces based on their defining exponents.

Findings

Surfaces are isomorphic iff their exponents are permutations of each other.

The classification is complete for surfaces defined by sums of three monomials plus one.

The result relies on analyzing the structure of these specific algebraic surfaces.

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.