Images of the Polar maps for Hypersurfaces

TL;DR

This paper investigates the images of polar maps of various degrees for projective hypersurfaces, computes their cohomology classes, and applies classical dual variety results to deepen understanding of hypersurface geometry.

Contribution

It introduces a detailed study of polar map images for hypersurfaces, including cohomology class calculations and applications to dual varieties, advancing classical algebraic geometry knowledge.

Findings

Cohomology classes of polar map images are explicitly calculated.

Classical dual variety results are extended and applied.

Insights into the geometry of hypersurfaces via polar maps are provided.

Abstract

For a projective hypersurface $X \subset \P^n$, the images of the polar maps of degree $k$ are studied. The cohomology class defined by these maps is calculated and classical results on dual varieties are presented as applications.