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Scheme-theoretic Whitney conditions

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Abstract

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.

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Acknowledgements

The author is very grateful to Christian Peskine and Fyodor Zak for the numerous and fruitful discussions they have had on Conjecture 1.2. It was their insight that the the linear span of the union of the tangent spaces to the tangency scheme should replace the linear span of the tangency scheme in the statement of Conjecture 1.2. The author also indebted to Claire Voisin for sharing with him her counter-example to Conjecture 1.2.

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  1. Rectorat de Paris, Département de mathématiques, 47 rue des Écoles, 75005, Paris, France

    Roland Abuaf

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  1. Roland Abuaf
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Abuaf, R. Scheme-theoretic Whitney conditions. European Journal of Mathematics 6, 249–261 (2020). https://doi.org/10.1007/s40879-018-00308-1

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