Fubini's theorem in codimension two

Abstract

We classify codimension two analytic submanifolds of projective space X ⊂ CP having the property that any line through a general point x ∈ X having contact to order two with X at x automatically has contact to order three. We give applications to the study of the Debarre-de Jong conjecture and of varieties whose Fano variety of lines has dimension 2n − 4.