Abstract algebraic variety
Encyclopedia
In algebraic geometry
, an abstract algebraic variety is an algebraic variety
that is defined intrinsically, that is, without an embedding
into another variety.
In classical algebraic geometry, all varieties were by definition quasiprojective varieties
, meaning that they were open subvarieties
of closed subvarieties of projective space
. In particular, they had a chosen embedding into projective space, and this embedding was used to define the topology on the variety and the regular function
s 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 P1×P1 is not a variety until it is embedded into the projective space; this is usually done by the Segre embedding
. However, any variety which admits one embedding into projective space admits many others by composing the embedding with the Veronese embedding. Consequently many notions which should be intrinsic, such as the concept of a regular function, are not obviously so.
The earliest successful attempt to define an abstract algebraic variety 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, it was Alexander Grothendieck
's definition of a scheme
that was both most general and found 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. Classical algebraic varieties are the quasiprojective integral separated finite type schemes over an algebraically closed field.
. Nagata's example was not complete (the analog of compactness), but soon afterwards he found an algebraic surface
which was complete and non-projective. Since then other examples have been found.
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, an abstract algebraic variety is an algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
that is defined intrinsically, that is, without an embedding
Embedding
In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....
into another variety.
In classical algebraic geometry, all varieties were by definition quasiprojective varieties
Quasiprojective variety
In mathematics, a quasiprojective variety in algebraic geometry is a locally closed subset of a projective variety, i.e., the intersection inside some projective space of a Zariski-open and a Zariski-closed subset...
, meaning that they were open subvarieties
Subvariety
In botanical nomenclature, a subvariety is a taxonomic rank below that of variety but above that of form : it is an infraspecific taxon. Its name consists of three parts: a genus name, a specific epithet and an infraspecific epithet. To indicate the rank, the abbreviation "subvar." should be put...
of closed subvarieties of projective space
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
. In particular, they had a chosen embedding into projective space, and this embedding was used to define the topology on the variety and the regular function
Regular function
In mathematics, a regular function is a function that is analytic and single-valued in a given region. In complex analysis, any complex regular function is known as a holomorphic function...
s 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 P1×P1 is not a variety until it is embedded into the projective space; this is usually done by the Segre embedding
Segre embedding
In mathematics, the Segre embedding is used in projective geometry to consider the cartesian product of two or more projective spaces as a projective variety...
. However, any variety which admits one embedding into projective space admits many others by composing the embedding with the Veronese embedding. Consequently many notions which should be intrinsic, such as the concept of a regular function, are not obviously so.
The earliest successful attempt to define an abstract algebraic variety was made by André Weil
André Weil
André Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...
. In his Foundations of Algebraic Geometry, Weil defined an abstract algebraic variety using valuations. Claude Chevalley
Claude Chevalley
Claude Chevalley was a French mathematician who made important contributions to number theory, algebraic geometry, class field theory, finite group theory, and the theory of algebraic groups...
made a definition of a scheme which served a similar purpose, but was more general. However, it was Alexander Grothendieck
Alexander Grothendieck
Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...
's definition of a scheme
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
that was both most general and found the most widespread acceptance. In Grothendieck's language, an abstract algebraic variety is usually defined to be an integral, separated scheme
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
of finite type over an algebraically closed field
Algebraically closed field
In mathematics, a field F is said to be algebraically closed if every polynomial with one variable of degree at least 1, with coefficients in F, has a root in F.-Examples:...
, although some authors drop the irreducibility or the reducedness or the separateness condition or allow the underlying field to be not algebraically closed. 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 NagataMasayoshi Nagata
Masayoshi Nagata was a Japanese mathematician, known for his work in the field of commutative algebra....
. Nagata's example was not complete (the analog of compactness), but soon afterwards he found an algebraic surface
Algebraic surface
In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two and so of dimension four as a smooth manifold.The theory of algebraic surfaces is much more complicated than that...
which was complete and non-projective. Since then other examples have been found.