Desingularisaatio

Desingularization, or resolution of singularities, is a fundamental concept in algebraic geometry and related fields. It refers to the process of transforming a possibly singular algebraic variety into a smooth one, in a way that preserves its essential birational properties. Concretely, for a variety X over a field k, a resolution consists of a smooth variety Y and a proper birational morphism f: Y → X that is an isomorphism over the smooth locus of X. In many contexts, one also considers embedded resolutions, where X sits inside a smooth ambient space W and the goal is to obtain a proper birational map from a smooth Z to W such that the strict transform of X is smooth and meets the exceptional divisors with simple normal crossings.

Historically, the cornerstone result is Hironaka’s theorem (1964), which established resolution of singularities for varieties over fields of characteristic zero. This landmark proof has inspired subsequent, more constructive approaches by Bierstone–Milman, Villamayor, and others, yielding algorithms that are canonical or depend on choices with controlled behavior. In positive characteristic, a complete general resolution is still open in high dimensions, though there are well-developed results in characteristic p for surfaces and several special cases. Moreover, significant progress has been made in embedded and log resolutions, which are central to many applications.

Key methods center on blow-ups, a type of birational modification that replaces a locus with its projectivized normal bundle, introducing exceptional divisors to improve singularities. Iterated sequences of blow-ups along smooth centers can produce a smooth variety and, in embedded settings, a divisor with simple normal crossings.

Desingularization is essential in birational geometry, moduli problems, integration on singular spaces, and algorithmic algebraic geometry, providing a bridge between singular models and smooth, better-understood counterparts.