Partial desingularization up to normal-crossings in characteristic 0 and 2

Abstract

We address the question of normal-crossings-preserving resolution of singularities (NC-preserving resolution), and compare the cases of characteristic 0 and characteristic 2. In characteristic 0, it is shown by Belotto da Silva and Bierstone arxiv:2602.09114 and W{\l}odarczyk arxiv:2602.14266 that, if one allows to introduce stack theoretic weighted blowups, any variety over a field of characteristic 0 admits a normal crossings resolution. We provide a principle that makes such results possible, Theorem 3.1.4. We further show that the coarse moduli space can be restricted to have higher pinch points (Definition 1.3.2), see Theorem 1.3.3. In contrast, in characteristic 2 we classify pinch points, and show that weighted blowups cannot lead to NC-preserving resolution of pinch points, although a pinch point is always the coarse moduli space of a NC stack (Theorem 1.4.1).