On Aluffi's problem and blowup algebras of certain modules

This paper deals with blowup algebras of certain classical modules related to a quasi-homogeneous hypersurface in characteristic zero. Motivated by Aluffi's problem, which asks for an appropriate adaptation of his quasi-symmetric algebra into the context of modules (or sheaves), we propose and study our general candidate - which is shown to actually retrieve Aluffi's definition in the situation of ideals - and further we compute it explicitly in the case of the module of tangent vector fields on the (possibly singular) given divisor. Along the way, we work out the Rees ring of the corresponding logarithmic derivation module and we apply a structural result of Corso-Polini-Ulrich in order to determine its core (intersection of all reductions) under certain conditions.