Normal Variety

Definition

A normal variety is a type of algebraic variety that satisfies the condition of normality, meaning it has no 'bad' points in terms of singularities or irregularities. Specifically, a variety is normal if every local ring of its coordinate ring is integrally closed, which means any element that is integral over that ring must already be in the ring itself. This property helps ensure that normal varieties behave well under various mathematical operations, especially in the context of birational equivalence and isomorphisms.

5 Must Know Facts For Your Next Test

1. Normal varieties are essential in algebraic geometry because they allow for better control over morphisms and maps between varieties.
2. A key theorem states that a projective variety over an algebraically closed field is normal if and only if it is irreducible and reduced.
3. Normality can often be checked locally, meaning one can analyze small neighborhoods around points to determine if the entire variety is normal.
4. Many familiar varieties, such as smooth varieties, are normal, while some varieties with singular points may fail to be normal.
5. The concept of normalization helps in constructing a normal variety from a given variety by resolving singularities and removing irregularities.

Review Questions

• How does the property of normality influence the study of morphisms between varieties?
  • Normality ensures that morphisms between varieties behave predictably and allows for better control over their properties. Since normal varieties have well-defined local rings that are integrally closed, this means that when considering maps between them, singularities and irregularities do not create complications that could disrupt the structure. This property is crucial in defining and analyzing rational maps, as it allows for the consistent application of algebraic techniques.
• Discuss how normalization relates to birational equivalence and why it is important in algebraic geometry.
  • Normalization plays a vital role in establishing birational equivalence between varieties by helping to identify when two varieties can be connected through rational maps. Normal varieties are often easier to work with because their integral closure ensures fewer singular points, facilitating the construction of birational models. This process helps mathematicians understand how different varieties can represent similar geometric structures despite potential differences in their definitions or forms.
• Evaluate the significance of normal varieties in the context of singularities and their resolution within algebraic geometry.
  • Normal varieties significantly contribute to the understanding and resolution of singularities within algebraic geometry. Since normality ensures that varieties do not possess problematic singular points, it aids in creating a clearer framework for studying how these singularities can be resolved. The resolution of singularities often involves transforming a non-normal variety into a normal one, which allows for a more comprehensive analysis of its geometric properties and behaviors. This transformation ultimately connects various aspects of algebraic geometry, linking the concepts of smoothness, normality, and birational relationships.

Related terms

Singular Variety: A singular variety is one that has points where the space fails to be smooth or well-behaved, often causing complications in the study of its geometric properties.

Birational Equivalence: Birational equivalence refers to a relationship between two varieties where they can be related by rational maps, indicating that they have similar geometric properties despite possible differences.

Integral Closure: Integral closure of a ring is the set of elements that are integral over the ring, playing a crucial role in defining normal varieties and ensuring they are well-behaved.