Higher order polar and reciprocal polar varieties

TL;DR

This paper introduces higher order polar and reciprocal polar varieties, extending classical concepts by using higher order osculating spaces, and explores their duality, degrees, and computational aspects for specific algebraic varieties.

Contribution

It generalizes classical polar loci to higher orders, establishing new duality relations and methods for computing degrees of these varieties in special cases.

Findings

Higher order polar loci are natural extensions of classical polar loci.

Degree of top polar class equals the degree of the corresponding higher order dual variety.

Explicit computations are provided for curves, scrolls, and toric varieties.

Abstract

In this note we introduce higher order polar loci as natural generalizations of the classical polar loci, replacing the role of tangent spaces by that of higher order osculating spaces. The close connection between polar loci and dual varieties carries over to a connection between higher order polar loci and higher order dual varieties. We generalize the duality between the degrees of polar classes of a variety and those of its dual variety to varieties that are reflexive to a higher order. In particular, the degree of the top (highest codimension) polar class of order k is equal to the degree of the k-th dual variety. We define higher order Euclidean normal bundles and use them to define higher order reciprocal polar loci and classes. We give examples of how to compute the degrees of the higher order polar and reciprocal polar classes in some special cases: curves, scrolls, and toric varieties.