Special geodesics and atypical intersections

TL;DR

This paper investigates complex plane curves, demonstrating that such curves contain finitely many real algebraic curves with projections composed of special geodesics, excluding those from modular polynomial vanishing loci.

Contribution

It establishes finiteness results for real algebraic curves with specific geodesic projections within complex irreducible plane curves, excluding modular polynomial cases.

Findings

Finitely many real algebraic curves with special geodesic projections exist on certain complex curves.

Excludes curves that are vanishing loci of modular polynomials.

Provides a classification related to special geodesics on complex curves.

Abstract

Let $C$ be a complex irreducible plane curve that is not the vanishing locus of a modular polynomial. We show that $C$ contains finitely many real algebraic curves whose projection on each coordinate axis is a union of special geodesics.