desingularización

Desingularización is a process in algebraic geometry used to eliminate singular points from an algebraic variety. Singularities are points where the variety is not "smooth," meaning it lacks a well-defined tangent space. These points can arise from various geometric phenomena like self-intersections or cusps. The goal of desingularización is to replace these singular points with a collection of smooth points in a way that preserves as much of the original variety's structure as possible.

The most common method for desingularización is the Hironaka theorem, which states that every algebraic variety over a field of characteristic zero can be desingularized. This process typically involves blowing up the variety at the singular points. Blowing up is an operation that replaces a point or a subvariety with a projective space, effectively stretching out the neighborhood of the singularity. By repeatedly applying this process, one can resolve all the singularities, resulting in a smooth variety, often called a resolution of singularities.

Desingularización is a fundamental tool in algebraic geometry with many applications. It allows mathematicians to study the properties of singular varieties by transforming them into smooth ones, which are generally easier to analyze. This process is crucial for understanding concepts like intersection theory, Hodge theory, and the classification of algebraic varieties. While Hironaka's theorem guarantees the existence of desingularizations, finding explicit ones can be computationally challenging.