Algebraic variety

Algebraic 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. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition.

Conventions regarding the definition of an algebraic variety differ slightly. 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. Under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility.

The fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial (an algebraic object) in one variable with complex number coefficients is determined by the set of its roots (a geometric object) in the complex plane. Generalizing this result, Hilbert's Nullstellensatz provides a fundamental correspondence between ideals of polynomial rings and algebraic sets. Using the ''Nullstellensatz'' and related results, mathematicians have established a strong correspondence between questions on algebraic sets and questions of ring theory. This correspondence is a defining feature of algebraic geometry.

Many algebraic varieties are manifolds, but an algebraic variety may have singular points while a manifold cannot. Algebraic varieties can be characterized by their dimension. Algebraic varieties of dimension one are called algebraic curves and algebraic varieties of dimension two are called algebraic surfaces.

In the context of modern scheme theory, an algebraic variety over a field is an integral (irreducible and reduced) scheme over that field whose structure morphism is separated and of finite type.

Overview and definitions

An ''affine variety'' over an algebraically closed field is conceptually the easiest type of variety to define, which will be done in this section. Next, one can define projective and quasi-projective varieties in a similar way. The most general definition of a variety is obtained by patching together smaller quasi-projective varieties. It is not obvious that one can construct genuinely new examples of varieties in this way, but Nagata gave an example of such a new variety in the 1950s.

Affine varieties

For an algebraically closed field and a natural number , let be an affine -space over , identified to $K^n$ through the choice of an affine coordinate system. The polynomials in the ring can be viewed as ''K''-valued functions on by evaluating at the points in , i.e. by choosing values in ''K'' for each ''x_i''. For each set ''S'' of polynomials in , define the zero-locus ''Z''(''S'') to be the set of points in on which the functions in ''S'' simultaneously vanish, that is to say

:$Z(S) = \left \.$

A subset ''V'' of is called an affine algebraic set if ''V'' = ''Z''(''S'') for some ''S''. A nonempty affine algebraic set ''V'' is called irreducible if it cannot be written as the union of two proper algebraic subsets. An irreducible affine algebraic set is also called an affine variety. (Many authors use the phrase ''affine variety'' to refer to any affine algebraic set, irreducible or not.Hartshorne, p.xv, notes that his choice is not conventional; see for example, Harris, p.3)

Affine varieties can be given a natural topology by declaring the closed sets to be precisely the affine algebraic sets. This topology is called the Zariski topology.

Given a subset ''V'' of , we define ''I''(''V'') to be the ideal of all polynomial functions vanishing on ''V'':

:$I(V) = \left \.$

For any affine algebraic set ''V'', the coordinate ring or structure ring of ''V'' is the quotient of the polynomial ring by this ideal.

Projective varieties and quasi-projective varieties

Let be an algebraically closed field and let be the projective ''n''-space over . Let in be a homogeneous polynomial of degree ''d''. It is not well-defined to evaluate on points in in homogeneous coordinates. However, because is homogeneous, meaning that , it ''does'' make sense to ask whether vanishes at a point . For each set ''S'' of homogeneous polynomials, define the zero-locus of ''S'' to be the set of points in on which the functions in ''S'' vanish:

:$Z(S) = \.$

A subset ''V'' of is called a projective algebraic set if ''V'' = ''Z''(''S'') for some ''S''. An irreducible projective algebraic set is called a projective variety.

Projective varieties are also equipped with the Zariski topology by declaring all algebraic sets to be closed.

Given a subset ''V'' of , let ''I''(''V'') be the ideal generated by all homogeneous polynomials vanishing on ''V''. For any projective algebraic set ''V'', the coordinate ring of ''V'' is the quotient of the polynomial ring by this ideal.

A quasi-projective variety is a Zariski open subset of a projective variety. Notice that every affine variety is quasi-projective. Notice also 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.

Abstract varieties

In classical algebraic geometry, all varieties were by definition quasi-projective varieties, meaning that they were open subvarieties of closed subvarieties of projective space. 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 (also called an abstract 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. So classically the definition of an algebraic variety required an embedding into projective space, and this embedding was used to define the topology on the variety and the regular functions on the variety. The disadvantage of such a definition is that not all varieties come with natural embeddings into projective space. For example, under this definition, the product is not a variety until it is embedded into the projective space; this is usually done by the Segre embedding. However, any variety that admits one embedding into projective space admits many others by composing the embedding with the Veronese embedding. Consequently, many notions that should be intrinsic, such as the concept of a regular function, are not obviously so.

The earliest successful attempt to define an algebraic variety abstractly, without an embedding, was made by André Weil. In his '' Foundations of Algebraic Geometry'', Weil defined an abstract algebraic variety using valuations. 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. 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.Liu, Qing. ''Algebraic Geometry and Arithmetic Curves'', p. 55 Definition 2.3.47, and p. 88 Example 3.2.3 Classical algebraic varieties are the quasiprojective integral separated finite type schemes over an algebraically closed field.

Existence of non-quasiprojective abstract algebraic varieties

One of the earliest examples of a non-quasiprojective algebraic variety were given by Nagata. Nagata's example was not complete (the analog of compactness), but soon afterwards he found an algebraic surface that was complete and non-projective. Since then other examples have been found; for example, it is straightforward to construct a toric variety that is not quasi-projective but complete.

Examples

Subvariety

A subvariety is a subset of a variety that is itself a variety (with respect to the structure induced from the ambient variety). For example, every open subset of a variety is a variety. See also