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 . ${\ displaystyle K}$

An affine algebraic set is a subset of an affine space that has the form ${\ displaystyle K ^ {n}}$

{\ displaystyle \ {x \ in K ^ {n} \ mid f_ {1} (x) = \ dotsb = f_ {k} (x) = 0 \}}

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.) ${\ displaystyle \ {f_ {1}, \ dotsc, f_ {k} \}}$ ${\ displaystyle K [X_ {1}, \ dotsc, X_ {n}]}$

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 : ${\ displaystyle Z \ subseteq K ^ {n}}$ ${\ displaystyle I (Z)}$ ${\ displaystyle Z}$

{\ displaystyle I (Z) = \ {f \ in K [X_ {1}, \ dotsc, X_ {n}] \ mid f (x) = 0 \ \ mathrm {f {\ ddot {u}} r \ all} \ x \ in Z \}}

The coordinate ring of an affine variety is the quotient ring ${\ displaystyle V}$

{\ displaystyle K [V]: = K [X_ {1}, \ dotsc, X_ {n}] / I (V)}

.

Such polynomials are identified with one another which match as a function on . ${\ displaystyle V}$

The quotient field of is the field of the rational functions . ${\ displaystyle K \ left [V \ right]}$ ${\ displaystyle K (V)}$

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 . ${\ displaystyle P ^ {n}}$ ${\ displaystyle n}$ ${\ displaystyle K}$ ${\ displaystyle f \ in K [X_ {0}, \ dotsc, X_ {n}]}$ ${\ displaystyle x = [x_ {0}: \ dotsc: x_ {n}]}$ ${\ displaystyle f (x_ {0}, \ dotsc, x_ {n}) = 0}$ ${\ displaystyle x}$

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

{\ displaystyle \ {x \ in P ^ {n} \ mid f_ {1} (x) = \ dotsb = f_ {k} (x) = 0 \}}

for homogeneous polynomials in has. ${\ displaystyle f_ {1}, \ dotsc, f_ {k}}$ ${\ displaystyle K [X_ {0}, \ dotsc, X_ {n}]}$

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 . ${\ displaystyle Z \ subseteq P ^ {n}}$ ${\ displaystyle I (Z)}$ ${\ displaystyle Z}$ ${\ displaystyle V}$ ${\ displaystyle K [X_ {0}, \ dotsc, X_ {n}] / I (Z)}$

Affinity variety morphisms

If there are affine varieties, then a map is a morphism from to if there is a polynomial map with . ${\ displaystyle V \ subset K ^ {m}, W \ subset K ^ {n}}$ ${\ displaystyle \ phi \ colon V \ rightarrow W}$ ${\ displaystyle V}$ ${\ displaystyle W}$ ${\ displaystyle \ Phi \ colon K ^ {m} \ rightarrow K ^ {n}}$ ${\ displaystyle \ Phi \ mid _ {V} = \ phi}$

A morphism is an isomorphism if there is a morphism with it . ${\ displaystyle \ phi}$ ${\ displaystyle \ psi \ colon W \ rightarrow V}$ ${\ displaystyle \ phi \ circ \ psi = \ mathrm {id} _ {W}, \ psi \ circ \ phi = \ mathrm {id} _ {V}}$

dimension

The Krull dimension of an algebraic variety is the largest number such that a chain of irreducible closed subsets of exists. ${\ displaystyle V}$ ${\ displaystyle n}$ ${\ displaystyle Z_ {0} \ subsetneq Z_ {1} \ dotsb \ subsetneq Z_ {n}}$ ${\ displaystyle V}$

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. ${\ displaystyle x}$

A non-singular variety with an actual birational morphism is called the resolution of the singularities of a variety . ${\ displaystyle V}$ ${\ displaystyle W}$ ${\ displaystyle f \ colon W \ rightarrow V}$

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

The structure of algebraic varieties

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

[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