Scheme-theoretic Whitney conditions and applications to tangency of projective varieties

We investigate a scheme-theoretic variant of Whitney condition a. If X is a projective variety and Y ⊂ X a subvariety, then X satisfies generically the scheme-theoretic Whitney condition a along Y provided that the projective dual of X is smooth. We give applications to tangency of projective varieties. For instance, if C ⊂ P 3 is a smooth space curve, we prove that a generic osculating plane to C is not tangent to C outside its osculating locus.