Recovery of Plane Curves from Branch Points

TL;DR

This paper develops methods to recover plane algebraic curves, specifically cubics and quartics, from their branch points, providing exact algorithms and counting real solutions using classical algebraic geometry.

Contribution

It introduces algorithms for recovering plane curves from branch points and determines the number of real solutions for cubics and quartics.

Findings

Determined the number of real solutions for cubic and quartic curves.

Provided exact algorithms for curve recovery from branch points.

Counted solutions using plane Hurwitz numbers 40 and 120.

Abstract

We recover plane curves from their branch points under projection onto a line. Our focus lies on cubics and quartics. These have 6 and 12 branch points respectively. The plane Hurwitz numbers 40 and 120 count the orbits of solutions. We determine the numbers of real solutions, and we present exact algorithms for recovery. Our approach relies on 150 years of beautiful algebraic geometry, from Clebsch to Vakil and beyond.