Scheme-theoretic Whitney conditions

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We investigate a scheme-theoretic variant of Whitney condition (a). If X is a projective variety over the field of complex numbers and \(Y \subset X\) 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 over \({\mathbb {C}}\) and to convex real algebraic geometry. In particular, we prove a Bertini-type theorem for osculating planes of smooth complex space curves and a generalization of a theorem of Ranestad and Sturmfels describing the algebraic boundary of an affine compact real variety.