Special Student Algebraic Geometry Seminar: Chow Classes of Varieties of Secant and Tangent Lines

Given a nondegenerate smooth variety X in P^n, let S(X) (resp. T(X)) be the subvariety of the Grassmannian Gr(2, n+1) consisting of secant (resp. tangent) lines to X. I will give closed-form formulae for the classes of S(X) and T(X) in the Chow ring of Gr(2,n+1) in terms of the “higher degrees” of the embedding, by a simple application of the Excess Intersection Formula on a flag variety. Using these formulae, one can recover classical results about the degree of the subvariety Sec(X) (resp. Tan(X)) of P^n swept out by the lines in S(X) (resp. T(X)), when it has the expected dimension. Finally, I will suggest potential extensions of these techniques to varieties of trisecant or bitangent lines. (Joint work with Hannah Larson.)