Algebraic variety

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.

Generalizing this result, Hilbert's Nullstellensatz provides a fundamental correspondence between ideals of polynomial rings and algebraic sets.

For an algebraically closed field K and a natural number n, let An be an affine n-space over K, identified to

The polynomials  f  in the ring K[x1, ..., xn] can be viewed as K-valued functions on An by evaluating  f  at the points in An, i.e. by choosing values in K for each xi.

[1]: 10 Projective varieties are also equipped with the Zariski topology by declaring all algebraic sets to be closed.

The disadvantage of such a definition is that not all varieties come with natural embeddings into projective space.

The earliest successful attempt to define an algebraic variety abstractly, without an embedding, was made by André Weil.

Claude Chevalley made a definition of a scheme, which served a similar purpose, but was more general.

However, Alexander Grothendieck's definition of a scheme is more general still and has received the most widespread acceptance.

Let k = C, and A2 be the two-dimensional affine space over C. Polynomials in the ring C[x, y] can be viewed as complex valued functions on A2 by evaluating at the points in A2.

Let k = C, and A2 be the two-dimensional affine space over C. Polynomials in the ring C[x, y] can be viewed as complex valued functions on A2 by evaluating at the points in A2.

One approach in this case is to check that the projection (x, y, z) → (x, y) is injective on the set of the solutions and that its image is an irreducible plane curve.

does not vanish is called the characteristic variety of M.[6] The notion plays an important role in the theory of D-modules.

A plane projective curve is the zero locus of an irreducible homogeneous polynomial in three indeterminates.

The Grassmannian variety Gn(V) is the set of all n-dimensional subspaces of V. It is a projective variety: it is embedded into a projective space via the Plücker embedding: where bi are any set of linearly independent vectors in V,

is the n-th exterior power of V, and the bracket [w] means the line spanned by the nonzero vector w. The Grassmannian variety comes with a natural vector bundle (or locally free sheaf in other terminology) called the tautological bundle, which is important in the study of characteristic classes such as Chern classes.

There are few ways to show this moduli has a structure of a possibly reducible algebraic variety; for example, one way is to use geometric invariant theory which ensures a set of isomorphism classes has a (reducible) quasi-projective variety structure.

, a not-necessarily-smooth complete curve with no terribly bad singularities and not-so-large automorphism group.

Historically a paper of Mumford and Deligne[10] introduced the notion of a stable curve to show

Here, there are the notions of stable and semistable vector bundles on a smooth complete curve

The theory of toric varieties (or torus embeddings) gives a way to compactify

It is not projective either, since there is a nonconstant regular function on X; namely, p. Another example of a non-affine non-projective variety is X = A2 − (0, 0) (cf.

To be able to do more — for example, to deal with varieties over fields that are not algebraically closed — some foundational changes are required.

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

This definition works over any field k. It allows you to glue affine varieties (along common open sets) without worrying whether the resulting object can be put into some projective space.

This also leads to difficulties since one can introduce somewhat pathological objects, e.g. an affine line with zero doubled.

(Strictly speaking, there is also a third condition, namely, that one needs only finitely many affine patches in the definition above.)

This is one of several generalizations of classical algebraic geometry that are built into Grothendieck's theory of schemes.

Allowing nilpotent elements in rings is related to keeping track of "multiplicities" in algebraic geometry.