Normal variety

In algebraic geometry , a branch of mathematics , normal varieties are algebraic varieties with only mild singularities .

The term was introduced by Oscar Zariski in connection with his purely algebraic, commutative algebra- based foundation of algebraic geometry and his work on the resolution of singularities .

definition

An algebraic variety , or more generally a scheme , is a normal variety or a normal scheme if the local ring of each point is completely closed in its quotient field . ${\displaystyle X}$ ${\displaystyle {\mathcal {O}}_{x}(X)}$ ${\displaystyle x\in X}$

(The term normal is explained by the fact that rings which are completely closed in their quotient field are also called normal .)

criteria

The curve is not normal because there is a finite birational morphism from A ¹ to the curve that is not an isomorphism. ${\displaystyle x^{3}-y^{2}=0}$ ${\displaystyle t\to (t^{2},t^{3})}$

The following criteria are equivalent to an algebraic variety being normal. ${\displaystyle X}$

• The ring of regular functions is completely closed in its quotient field. ${\displaystyle {\mathcal {O}}(X)}$
• Every finite birational mapping from an algebraic variety to is an isomorphism . ${\displaystyle f\colon Y\to X}$ ${\displaystyle Y}$ ${\displaystyle X}$

properties

• Any regular scheme , i.e. H. any scheme without singularities is normal. Conversely, a normal variety only has singularities of codimension at least 2. In particular, the singularities of an algebraic curve are not normal.
• For a normal variety over (in classical topology as a subset of ) the link of each point is contiguous, i.e. H. each point has arbitrarily small neighborhoods so connected is. ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {C} ^{n}}$ ${\displaystyle U}$ ${\displaystyle U\setminus \left\{x\right\}}$

literature

• Oscar Zariski: Some Results in the Arithmetic Theory of Algebraic Varieties. , Amer. J. Math., 61 (2), 249-294, 1939.