A classical method to study a projective variety is to consider its hyperplane section and "lift" the properties of the section to the variety. This is sometimes called Apollonius method and it works well since in general a variety is at least as special as any of its hyperplane sections. For example a weighted projective space can be an hyperplane section only of a weighted projective space (S. Mori 1975).

 

We extend this result in a "relative situation", namely we consider to be a local, projective, divisorial contraction between normal varieties of dimension with -factorial singularities and  to be a -ample Cartier divisor. If has a structure of a weighted blow-up then , as well, has a structure of weighted blow-up.

 

As an application we consider a local projective contraction from a variety with terminal -factorial singularities, which contracts a prime divisor to an isolated -factorial singularity , such that \\

is -ample, for a -ample Cartier divisor on .

 

Using the above result, the existence of a "good" general section of and the existing results in dimension , we prove that is a hyperquotient singularity and is a weighted blow-up.