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.

Is an affine space an affine variety?

In algebraic geometry, an affine variety (or, more generally, an affine algebraic set) is defined as the subset of an affine space that is the set of the common zeros of a set of so-called polynomial functions over the affine space.

Is an affine variety irreducible?

Affine variety It is irreducible, as it cannot be written as the union of two proper algebraic subsets.

Is every variety quasi projective?

2.9 given by projective closure is in fact an isomorphism in the sense of section I. 3, so all varieties are isomorphic to quasi-projective varieties.

Are projective varieties compact?

Projective varieties form a very large class of “compact” varieties that do admit such a global de- scription. 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 — in this class we will certainly not see one.

What is an affine cone?

From Wikipedia, the free encyclopedia. In algebraic geometry, a cone is a generalization of a vector bundle. Specifically, given a scheme X, the relative Spec. of a quasi-coherent graded OX-algebra R is called the cone or affine cone of R. Similarly, the relative Proj.

What is affine hyperplane?

An affine hyperplane is an affine subspace of codimension 1 in an affine space. In Cartesian coordinates, such a hyperplane can be described with a single linear equation of the following form (where at least one of the ‘s is non-zero and is an arbitrary constant):

What is an affine subset?

An affine subset is defined (in Linear Algebra Done Right 3th edition) as a subset of vector space V, that can be expressed as v+U, where v∈V, U is a subspace of V.

Are varieties irreducible?

Definition An affine variety is reducible if it is the union of proper subvarieties . Otherwise, is irreducible. That is, an affine variety is irreducible if whenever is written in the form , where and are affine varieties, then either or .

How do you prove a variety is irreducible?

A variety V is irreducible if it can not be decomposed as V = V1 ∪ V2 where V1,V2 ⊊ V are strictly smaller varieties. An ideal I is prime if for every pair f,g ∈ R, fg ∈ I =⇒ f ∈ I or g ∈ I.

What is a quasi affine variety?

If the condition of generating a prime ideal is removed, such a set is called an (affine) algebraic set. 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.

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.