Logarithmic resolution of singularities via multi-weighted blow-ups
We revisit the theorem of Hironaka that one can resolve the singularities of a singular, reduced closed subscheme X of a smooth scheme Y over a field of characteristic zero, such that the singular locus of X is transformed to a simple normal crossings divisor. We propose a computable yet efficient algorithm, which accomplishes this by taking successive proper transforms along a sequence of multi-weighted blow-ups, where at each step, the worst singular locus is blown up, and one witnesses an immediate improvement in singularities. Here, multi-weighted blow-ups are necessary to ensure that the ambient space remains smooth (in fact, also logarithmically smooth with respect to the logarithmic structure associated to the exceptional divisors), although one has to work more broadly with Artin stacks. This is joint work with Dan Abramovich.