Glattpunkte

Glattpunkte are a concept in algebraic geometry, specifically related to the study of schemes. In essence, a Glattpunkt is a point in a scheme that has certain "niceness" properties. More formally, a scheme X is said to have Glattpunkte if it is locally of finite type over a field K. This means that every point in X has a neighborhood that can be represented as the spectrum of a finitely generated algebra over K. This property is crucial for many constructions and theorems in algebraic geometry, as it ensures that the scheme is "well-behaved" in a certain sense.

The term "Glattpunkt" is German for "smooth point." While the term is often used synonymously with "smooth point" in the context of smooth manifolds, in algebraic geometry, a Glattpunkt is a more technical definition. A Glattpunkt is a point where the tangent space has the minimal possible dimension, which is related to the dimension of the scheme itself. This condition implies that the local ring at that point is regular. Regular local rings are very well-understood and possess many desirable properties, such as being a unique factorization domain.

The existence of Glattpunkte is important for understanding the geometry of a scheme. For example, a scheme that is of finite type over a field is guaranteed to have Glattpunkte everywhere except possibly at a finite number of points. These "singular" points are where the scheme deviates from being smooth. Studying the singular points is often a key part of understanding the overall structure and properties of an algebraic variety.