On the polar degree of projective hypersurfaces

TL;DR

This paper provides algebraic formulas for the degree of the polar map of hypersurfaces in complex projective space and classifies plane curves with low-degree polar maps, including homaloidal curves.

Contribution

It introduces purely algebraic methods to compute polar map degrees and extends formulas to hypersurface components, offering new insights into plane curve classifications.

Findings

Formulas for polar map degrees in terms of hypersurface components

Classification of plane curves with low-degree polar maps

Simplified proof of Dolgachev's classification of homaloidal curves

Abstract

Given a hypersurface in the complex projective $n$-space we prove several known formulas for the degree of its polar map by purely algebro-geometric methods. Furthermore, we give formulas for the degree of its polar map in terms of the degrees of the polar maps of its components. As an application, we classify the plane curves with polar map of low degree, including a very simple proof of I. Dolgachev's classification of homaloidal plane curves.