20 Facts About Algebraic variety


Algebraic variety varieties are the central objects of study in algebraic geometry, a sub-field of mathematics.

FactSnippet No. 1,513,218

Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers.

FactSnippet No. 1,513,219

For example, some definitions require an algebraic variety to be irreducible, which means that it is not the union of two smaller sets that are closed in the Zariski topology.

FactSnippet No. 1,513,220

The most general definition of a Algebraic variety is obtained by patching together smaller quasi-projective varieties.

FactSnippet No. 1,513,221

Quasi-projective Algebraic variety is a Zariski open subset of a projective Algebraic variety.

FactSnippet No. 1,513,222

Notice that the complement of an algebraic set in an affine variety is a quasi-projective variety; in the context of affine varieties, such a quasi-projective variety is usually not called a variety but a constructible set.

FactSnippet No. 1,513,223

For example, in Chapter 1 of Hartshorne a variety over an algebraically closed field is defined to be a quasi-projective variety, but from Chapter 2 onwards, the term variety refers to a more general object, which locally is a quasi-projective variety, but when viewed as a whole is not necessarily quasi-projective; i e it might not have an embedding into projective space.

FactSnippet No. 1,513,224

However, any Algebraic variety that admits one embedding into projective space admits many others by composing the embedding with the Veronese embedding.

FactSnippet No. 1,513,225

In Grothendieck's language, an abstract algebraic variety is usually defined to be an integral, separated scheme of finite type over an algebraically closed field, although some authors drop the irreducibility or the reducedness or the separateness condition or allow the underlying field to be not algebraically closed.

FactSnippet No. 1,513,226

Hilbert's Nullstellensatz says that closed subvarieties of an affine or projective Algebraic variety are in one-to-one correspondence with the prime ideals or homogeneous prime ideals of the coordinate ring of the Algebraic variety.

FactSnippet No. 1,513,227

General linear group is an example of a linear algebraic group, an affine variety that has a structure of a group in such a way the group operations are morphism of varieties.

FactSnippet No. 1,513,228

Projective Algebraic variety is a closed subAlgebraic variety of a projective space.

FactSnippet No. 1,513,229

The Grassmannian variety Gn is the set of all n-dimensional subspaces of V It is a projective variety: it is embedded into a projective space via the Plucker embedding:.

FactSnippet No. 1,513,230

Grassmannian Algebraic variety comes with a natural vector bundle called the tautological bundle, which is important in the study of characteristic classes such as Chern classes.

FactSnippet No. 1,513,231

The modern notion of a variety is considerably more abstract than the one above, though equivalent in the case of varieties over algebraically closed fields.

FactSnippet No. 1,513,232

An abstract algebraic variety is a particular kind of scheme; the generalization to schemes on the geometric side enables an extension of the correspondence described above to a wider class of rings.

FactSnippet No. 1,513,233

Basically, a Algebraic variety over is a scheme whose structure sheaf is a sheaf of -algebras with the property that the rings R that occur above are all integral domains and are all finitely generated -algebras, that is to say, they are quotients of polynomial algebras by prime ideals.

FactSnippet No. 1,513,234

Some modern researchers remove the restriction on a Algebraic variety having integral domain affine charts, and when speaking of a Algebraic variety only require that the affine charts have trivial nilradical.

FactSnippet No. 1,513,235

Complete Algebraic variety is a Algebraic variety such that any map from an open subset of a nonsingular curve into it can be extended uniquely to the whole curve.

FactSnippet No. 1,513,236

An algebraic manifold is an algebraic variety that is an m-dimensional manifold, and hence every sufficiently small local patch is isomorphic to k Equivalently, the variety is smooth .

FactSnippet No. 1,513,237