Duality of Numerical Characters of Polar Loci

Abstract

Let XaP be an r-dlmensional projective variety. For every integer k with Og/cgr consider an (n — r + k — 2)-dimensional linear subspace L(/c) of P . Since the dimension of the tangent space of X at a smooth point x e X is equal to r, the tangent space necessarily intersects L(&) in a space of at least k — 2 dimensions. The closure of the set of all smooth points of X where this intersection space has dimension at least k— 1 is called the /c-th polar locus of X and is denoted by M(L(fc)), following Piene [4]. For a general L(fc)5 every component of M(L(k)) has codimension fc in X and moreover the rational equivalence class of the cycle defined by M(L(/{)) is independent of L(fe). We denote this equivalence class by [Mfc] = [M(L(fc))] and call it the /c-th polar class. The degree nk of [Mfe] is called the /c-th class. The number /,*0 is nothing but the degree of X. We set ^fe = 0 for any integer k with /c r. On the other hand, the dual variety Y in the dual projective space F is defined as the closure of the set of tangent hyperplanes. A hyperplane is tangent to X if it contains the embedded tangent space of X at a smooth point x e X.