No inference to general resolution of singularities

The results of this page prove resolution for reduced projective plane curves by point blowups (Resolution of reduced plane curves by point blowups and the delta recurrence) and the regularity of point blowups of regular surfaces; the blowups used are proper and, in the point case, have the explicit projective chart descriptions of Blowups of finite type ideals are locally H-projective, and proper. These statements do not imply resolution of singularities for arbitrary varieties, nor for surfaces over imperfect fields with smoothness assertions about the resulting components, nor for schemes of dimension at least three. The termination argument for the curve case uses two features special to curves on surfaces: the one-dimensional normalization defect ${\delta }_k$, which decreases by $rm(m-1)\mathrm{/}2$ at each singular point blowup, and the pairwise contact order of two regular branches at a point of a regular surface, which decreases by one under an appropriate point blowup. In higher dimension there is no such defect count, and the embedded normal-crossing support produced for curves does not control the singularities of a general ambient scheme. The page therefore claims only the plane-curve resolution and the regularity statements stated and proved in its items, and the normal-crossing conclusion is asserted over the residual residue fields, not as smoothness over an imperfect base field.