Varieties with one apparent double point

The number of apparent double points of a smooth, irreducible projective variety $X$ of dimension $n$ in ${\text{Proj}}^{2n+1}$ is the number of secant lines to $X$ passing through the general point of ${\text{Proj}}^{2n+1}$. This classical notion dates back to Severi. In the present paper we classify smooth varieties of dimension at most three having one apparent double point. The techniques developed for this purpose allow to treat a wider class of projective varieties.