Partial Local Resolution by Characteristic Zero Methods

Abstract

We discuss to what extent the local techniques of resolution of singularities over fields of characteristic zero can be applied to improve singularities in general. For certain interesting classes of singularities, this leads to an embedded resolution via blowing ups in regular centers. We illustrate this for generic determinantal varieties. The article is partially expository and is addressed to non-experts who aim to construct resolutions for other special classes of singularities in positive or mixed characteristic. A first approximation for the singularity of E = (J, b) at M is given by the tangent cone pair T M (E) = (In M (J, b), b) (Definition 1.23). It is a pair on the associated graded ring of R, gr M (R) := i≥0 M i /M i+1 (which is isomorphic to a polynomial ring over the residue field), and In M (J, b) is a homogeneous ideal generated by the b-initial forms of elements of J (Definition 1.16). From this, we obtain a cone C := Spec(gr M (R)/In M (J, b)).Associated to C there is the directrix Dir M (E) of E (at M ). This is the largest sub-vector space V of Spec(gr M (R)) leaving the cone C stable under translation, C + V = C (Defintion