**MSC:**- 14M99 None of the above but in this section
- 53A07 Higher-dimension and -codimension surfaces in Euclidean

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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