Focal Loci of Algebraic Varieties I

Fabrizio Catanese; Cecilia Trifogli

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Focal Loci of Algebraic Varieties I

Fabrizio Catanese·Cecilia Trifogli

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Abstract

The focal locus $Σ_X$ of an affine variety $X$ is roughly speaking the (projective) closure of the set of points $O$ for which there is a smooth point $x \in X$ and a circle with centre $O$ passing through $x$ which osculates $X$ in $x$. Algebraic geometry interprets the focal locus as the branching locus of the endpoint map $ε$ between the Euclidean normal bundle $N_X$ and the projective ambient space ($ε$ sends the normal vector $O-x$ to its endpoint $O$), and in this paper we address two general problems : 1) Characterize the "degenerate" case where the focal locus is not a hypersurface 2) Calculate, in the case where $Σ_X$ is a hypersurface, its degree (with multiplicity)

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Fabrizio Catanese, Cecilia Trifogli. 2000-05-10. Focal Loci of Algebraic Varieties I. https://arxiv.org/abs/math/0005096

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