What is a variety algebraic geometry?
A variety is the set of common zeros to a collection of polynomials. In classical algebraic geometry, the polynomials have complex numbers for coefficients. Because of the fundamental theorem of algebra, such polynomials always have zeros.
Is Z an affine variety?
In particular, Z is not an affine variety. 5. Let V ⊂ kn and W ⊂ km be affine varieties.
What is a quasi affine variety?
A quasi-affine variety is an open subvariety of an affine algebraic variety. (As an open subspace of a Noetherian space it is automatically compact.) An example of a quasi-affine variety that is not affine is C2∖{(0,0)}.
Are projective varieties affine?
Affine varieties are almost never complete (or projective). In fact, a projective subvariety of an affine variety must have dimension zero. This is because only the constants are globally regular functions on a projective variety.
Are varieties schemes?
A variety is a scheme X over k such that X is integral and the structure morphism X \to \mathop{\mathrm{Spec}}(k) is separated and of finite type. This definition has the following drawback. Suppose that k’/k is an extension of fields.
What are coordinate rings?
Definition. Given an affine variety , the coordinate ring of is donoted by k[V] and defined to be the set of polynomials , where . The ring k[V] is often described as “the ring of polynomial functions on ”.
What do you mean by polynomial rings?
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field.
Is projective space a variety?
This is a generalization to every ground field of the compactness of the real and complex projective space. A projective space is itself a projective variety, being the set of zeros of the zero polynomial.
Are projective varieties compact?
Projective varieties form a large class of “compact” varieties that do admit such a unified global description. In fact, the class of projective varieties is so large that it is not easy to construct a variety that is not (an open subset of) a projective variety.
What is a K scheme?
An algebraic k-scheme is a scheme X over k such that the structure morphism X \to \mathop{\mathrm{Spec}}(k) is of finite type. A locally algebraic k-scheme is a scheme X over k such that the structure morphism X \to \mathop{\mathrm{Spec}}(k) is locally of finite type.
What is affine variety in Algebra?
Affine variety. In algebraic geometry, an affine variety, or affine algebraic variety, over a algebraically closed field k is the zero-locus in k n {displaystyle k^{n}} of some finite family of polynomials of n variables with coefficients in k that generate a prime ideal.
What is a regular map of affine varieties?
A morphism, or regular map, of affine varieties is a function between affine varieties which is polynomial in each coordinate: more precisely, for affine varieties V ⊆ kn and W ⊆ km, a morphism from V to W is a map φ : V → W of the form φ(a1., an) = (f1(a1., an)., fm(a1., an)), where fi ∈ k[X1., Xn] for each i = 1., m.
What is a quasi-affine variety?
A Zariski open subvariety of an affine variety is called a quasi-affine variety . Some texts do not require a prime ideal, and call irreducible an algebraic variety defined by a prime ideal. This article refers to zero-loci of not necessarily prime ideals as affine algebraic sets .
What is the normalization of irreducible affine varieties?
The normalization of an irreducible affine variety is affine; the coordinate ring of the normalization is the integral closure of the coordinate ring of the variety. (Similarly, the normalization of a projective variety is a projective variety.)