desingularisaation

Desingularisation, also spelled desingularization (or desingularisation in British English), is a central concept in algebraic geometry. It refers to the process or result of replacing a possibly singular algebraic variety by a non-singular one through a morphism that is birational and an isomorphism over the smooth locus. Concretely, a desingularisation of a variety X over a field k is a proper birational morphism π: X′ → X with X′ smooth over k, such that π induces an isomorphism on the complement of the singular locus of X.

A fundamental result, due to Hironaka, states that every variety over a field of characteristic zero admits a desingularisation; equivalently, there exists a smooth X′ and a birational morphism to X. In positive characteristic, resolution is known in many cases (for example for curves and certain classes of higher-dimensional varieties) but remains open in full generality. In practice one often speaks of embedded resolution: if X is embedded in a smooth ambient variety, one seeks a birational morphism that resolves X and yields a divisor with simple normal crossings.

The standard method uses a sequence of blow-ups along smooth centers chosen to progressively improve the singularities. Over recent decades, algorithms have been developed to make the process constructive and, in favorable settings, functorial with respect to morphisms; notable approaches include those of Hironaka, Bierstone–Milman, Villamayor, and Włodarczyk.

Desingularisation has wide-ranging applications in birational geometry, Hodge theory, and the study of invariants on singular spaces. It provides a means to extend theories from smooth varieties to singular ones by passing to a smooth model.