Lifting from an ample section.The case of weighted blow-ups

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 sometime called Aplollonius 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 f : X → Z to be a local, projective, divisorial contraction between normal varieties of dimension n with Q-factorial singularities and Y ⊂ X to be a f-ample Cartier divisor. If f|Y : Y → W has a structure of a weighted blow-up then f : X → Z, as well, has a structure of weighted blow-up. As an application we consider a local projective contraction f : X → Z from a variety X with terminal Q-factorial singularities, which contracts a prime divisor E to an isolated Q-factorial singularity P ∈ Z, such that −(KX + (n − 3)L) is f-ample, for a f-ample Cartier divisor L on X. Using the above result, the existence of a ”good” general section of L and the existing results in dimension 3, we prove that (Z,P) is a hyperquotient singularity and f is a weighted blow-up.