Abstract
The focal locus ∑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 inx. Algebraic geometry interprets the focal locus as the branching locus of the endpoint map ∈ between the Euclidean normal bundle Nx and the projective ambient space (∈ 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 hyper surface. 2)Calculate, in the case where ∑x is a hypersurface, its degree (with multiplicity).
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