Proof of Schubert's conjectures on double contacts.

Abstract

The purpose of the paper under review is to give a proof of six formulas by Schubert (two of which he proved and four of which he only conjectured) concerning the number of double contacts among the curves of two families of plane curves. The method consists in finding bases of the Chow groups of the Hilbert scheme of length 2 subschemes of the point- line incidence variety. This approach turns out to be much simpler than the one using the space of triangles as suggested by Schubert. As a byproduct, the authors obtain proofs of the classical formulas on triple contacts (i.e., single contacts of third order) between two such families of curves