Desingularization

Desingularization, also known as resolution of singularities, is the process of replacing a singular algebraic variety by a nonsingular one that is birationally equivalent. This is typically achieved by a sequence of blowups along smooth centers, producing a smooth variety and an exceptional divisor that records the history of the transformation. In the embedded version, one resolves a subvariety inside a smooth ambient space, obtaining a nonsingular strict transform together with an easily controlled exceptional locus.

In characteristic zero, a foundational result is Hironaka’s theorem, which asserts that every algebraic variety over a field of characteristic zero admits a resolution of singularities. The method uses invariant-based selection of blowup centers to iteratively reduce a well-founded measure, such as multiplicity or a Hilbert-Samuel function. The outcome is a smooth variety, often accompanied by a divisor with simple normal crossings that reflects the combinatorics of the exceptional loci.

Canonical and algorithmic approaches have been developed to make desingularization explicit and functorial. Notable frameworks include the Bierstone–Milman and Villamayor procedures, as well as subsequent refinements by Włodarczyk. These algorithms aim for (weak) functoriality with respect to smooth morphisms and frequently address the related problem of principalization of ideals, transforming a given ideal into a principal, locally invertible divisor via blowups.

Limitations and scope: In positive or mixed characteristic, a general resolution is not known in full generality, though resolutions are known in many cases (low dimensions or special classes of varieties). Desingularization remains a central tool in birational geometry, moduli problems, cohomology theories, and the study of degenerations, providing a controlled way to work with singular spaces.