On rational varieties smooth except at a seminormal singular point

We construct a class of projective rational varieties X of any dimension m >= 1 which are smooth except at a point O, with the projective space P^m as normalization, having smooth branches and reduced projectivized tangent cone in O. The Hilbert function of X is considered and is explicitly computed when the point O is seminormal. Indeed, we study seminormality, obtaining necessary and sucient conditions for O to be seminormal and show that in such case the tangent cone is reduced and seminormal.