Focal Loci of Algebraic Varieties I

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)