# 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}$ ${\ displaystyle f \ colon W \ rightarrow V}$