Desingularisation

Desingularisation is a process in algebraic geometry and related fields that aims to resolve singularities of a geometric object, typically an algebraic variety or a scheme. Singularities are points where the object is not "smooth" in a geometric sense, meaning it might have sharp points, self-intersections, or other irregularities. These singularities can make it difficult to study the object's properties using standard calculus-based methods.

The goal of desingularisation is to replace the singular object with a smooth one, or at least a "better behaved" one, in a way that preserves as much of the original object's structure as possible. This is often achieved by introducing new points or curves, which are called "exceptional divisors" or "strict transforms," that "smooth out" the problematic regions.

A key result in this area is Hironaka's theorem, which states that any algebraic variety over a field of characteristic zero can be desingularised. This means that for such varieties, there always exists a sequence of operations that will eliminate all singularities. The process can involve techniques like blowing up points or subvarieties, which effectively replaces a singular point with a copy of a lower-dimensional projective space.

Desingularisation is a fundamental tool for understanding the geometry of singular varieties. It allows mathematicians to apply powerful techniques that are only valid for smooth objects, leading to deeper insights into their structure, topology, and classification. The study of desingularisation is crucial for many areas of modern mathematics, including differential geometry, complex analysis, and theoretical physics.