A natural Chow lemma for smooth threefolds and fourfolds

Abstract

Let $X$ be a smooth proper integral variety of dimension at most four over a field of characteristic zero. We construct a finite sequence of blowups along smooth centers with projective endpoint, preserving the intersection of all maximal quasi-projective opens of $X$. The construction depends only on a fixed functorial principalization procedure and is natural under isomorphisms, including those over different ground fields. The proof has two parts. A colength argument on regular local surfaces gives, in every dimension, a finite preparation after which some maximal quasi-projective open has complement mapping into codimension at least three in $X$. In dimensions at most four these images are points or curves. Ample numerical classes on the normalized components of that complement give finite bounds for summing base ideals of linear systems. The resulting ideal descends to the original field; geometric integrality is not required.