Algebraic variety

from Wikipedia, the free encyclopedia

In classical algebraic geometry , a branch of mathematics , an algebraic variety is a geometric object that can be described by polynomial equations .


Affine varieties

Let it be a solid, algebraically closed body .

An affine algebraic set is a subset of an affine space that has the form

for a (finite) set of polynomials in has. ( Hilbert's basic theorem states that any infinite system of polynomial equations can be replaced by an equivalent one with a finite number of equations.)

An affine variety is an irreducible affine algebraic set; H. a nonempty algebraic set that is not the union of two proper algebraic subsets.

The algebraic subsets of an affine variety can be understood as closed sets of a topology , the Zariski topology . A quasi-affine variety is an open subset of an affine variety.

For a set, let the vanishing ideal , i.e. the ideal of all polynomials that vanish entirely :

The coordinate ring of an affine variety is the quotient ring


Such polynomials are identified with one another which match as a function on .

The quotient field of is the field of the rational functions .

Projective varieties

In some contexts, affine varieties do not behave well because “points at infinity” are missing. Projective varieties, on the other hand, are complete . This fact is reflected, for example, in Bézout's theorem , which provides an exact formula for the number of points of intersection of projective plane curves, but only an estimate for affine plane curves.

Let it be the -dimensional projective space above the body . For a homogeneous polynomial and a point , the condition is independent of the chosen homogeneous coordinates of .

A projective algebraic set is a subset of projective space that has the form

for homogeneous polynomials in has.

A projective variety is an irreducible projective algebraic set.

The Zariski topology is also defined on projective varieties in such a way that the closed sets are exactly the algebraic subsets. A quasi-projective variety is an open subset of a projective variety.

For a projective algebraic set, let the vanishing ideal be, i.e. the ideal that is generated by the homogeneous polynomials that vanish completely . The homogeneous coordinate ring of a projective variety is the quotient ring .

Affinity variety morphisms

If there are affine varieties, then a map is a morphism from to if there is a polynomial map with .

A morphism is an isomorphism if there is a morphism with it .


The Krull dimension of an algebraic variety is the largest number such that a chain of irreducible closed subsets of exists.

The dimension of an affine variety is equal to the dimension of its coordinate ring. The dimension of a projective variety is one less than the dimension of its homogeneous coordinate ring.


A point of an algebraic variety or, more generally, of a schema is called singular (or: is a singularity ) if the associated local ring is not regular . For closed points of algebraic varieties this is equivalent to the fact that the dimension of the Zariski tangent space is larger than the dimension of the variety.

A non-singular variety with an actual birational morphism is called the resolution of the singularities of a variety .


Web links

Individual evidence

  1. Definition e.g. B. Hartshorne Algebraic Geometry . However, some authors dispense with the definition of irreducible , e.g. B. Hazewinkel Encyclopedia of Mathematics , Springer Online Reference . See also Eisenbud Commutative Algebra with applications to algebraic geometry , Springer, p. 32