Fabrizio Catanese, Cecilia Trifogli
Focal Loci of Algebraic Varieties I
The paper is published: Mathematica Gottingensis
MSC:
14M99 None of the above but in this section
53A07 Higher-dimension and -codimension surfaces in Euclidean
Abstract: \\
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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Keywords: Focal loci, Focal Hypersurfaces, Euclidean Geometry, evolute, centres of curvatures, focal degeneracys
Notes: To appear in Comm. in Alg. ( volume in honour of Hartshorne's 60-th birthday)