Focal Loci of Algebraic Varieties I

Journal article

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)

Authors: Catanese, F; Trifogli, C

• Publication status: Published
• Journal: COMMUNICATIONS IN ALGEBRA
• Volume: 28
• Issue: 12
• Pages: 6017-6057
• Publication date: 2000-05-10
• DOI: 10.1080/00927870008827202
• EISSN: 1532-4125
• ISSN: 0092-7872
• Language: English
• Keywords: 14N15;14M99;53A07;53A20; math.DG; math.AG