20 Facts About Algebraic variety

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

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

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.

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

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

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.

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.

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

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.

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.

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.

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

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:.

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.

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.

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.

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.

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.

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.

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 .