Focal Loci of Algebraic Varieties I

2000/05/10 by Fabrizio Catanese, Catanese, Fabrizio, Cecilia Trifogli +1 · 2 citations
Computer Science · Engineering · Mathematics · #14M99 #14N15 #53A07 #53A20 #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Polynomial and algebraic computation #math.AG #math.DG #msc:14M99 #msc:14N15 #msc:53A07 #msc:53A20

paper · pdf · doi:10.48550/arxiv.math/0005096

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

arxiv created 2000/05/10 · openalex publication_date 2000/05/10 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ∈ 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)

Cited by

• The Hermitian Distance degree of an Algebraic Variety
• Infinite Euclidean Distance Discriminants

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