Focal Loci of Algebraic Varieties I

Details Focal Loci of Algebraic Varieties I Focal Loci of Algebraic Varieties I

2000-05-10

arxiv.org/abs/math/0005096v1

Mathematics

Algebraic Geometry

45 pages. To appear in Comm. in Alg. (volume in honour of Hartshorne's 60-th birthday)

Scientific paper

The focal locus $\Sigma_X$ 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 $\epsilon$ between the Euclidean normal bundle $N_X$ and the projective ambient space ($\epsilon$ 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 $\Sigma_X$ is a hypersurface, its degree (with multiplicity)

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