How to Reconstruct a Planar Map From its Branching Curve

TL;DR

This paper introduces an algorithm to reconstruct a planar map from its branching curve by utilizing the ramification curve as a key intermediate, advancing the understanding of algebraic map reconstruction.

Contribution

The paper presents a novel algorithm that reconstructs a planar map from its branching curve using the linear normalization of the ramification curve.

Findings

Successfully reconstructs maps from branching curves

Utilizes linear normalization of the ramification curve

Provides a new approach to algebraic map reconstruction

Abstract

We present an algorithm for constructing a map $\mathbb{P}^2\to\mathbb{P}^2$ with a given branching curve. The stepping stone is the ramification curve, which is obtained as the linear normalization of the branching curve.