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
Canonical and algorithmic approaches have been developed to make desingularization explicit and functorial. Notable frameworks include
Limitations and scope: In positive or mixed characteristic, a general resolution is not known in full generality,