On the three-term curves

TL;DR

This paper classifies certain smooth projective plane curves with a specific three-term form, describes their Belyi functions, and identifies their birational families, contributing to algebraic geometry and number theory.

Contribution

It provides a classification of three-term curves, details their Belyi functions, and explores their birational families, expanding understanding of these special algebraic curves.

Findings

Existence of three-term curves in any degree

Identification of five birational families

Explicit Belyi functions for these curves

Abstract

We classify projective plane nonsingular curves admitting a 3-term presentation; they exist in any degree, generally constitute 5 birational families and are defined over rational numbers. The Belyi functions on all these curves are presented.