Algebraic variety
In classical algebraic geometry , a branch of mathematics , an algebraic variety is a geometric object that can be described by polynomial equations .
Definitions
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 .
dimension
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.
Singularities
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 .
literature
- Klaus Hulek : Elementary Algebraic Geometry . Vieweg, Braunschweig / Wiesbaden 2000, ISBN 3-528-03156-5 .
- Robin Hartshorne : Algebraic Geometry . Springer-Verlag, New York 1977, ISBN 0-387-90244-9 .
Web links
Individual evidence
- ↑ 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