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 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 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,[1] although some authors drop the irreducibility or the reducedness or the separateness condition or allow the underlying field to be not algebraically closed.[2] Classical algebraic varieties are the quasiprojective integral separated finite type schemes over an algebraically closed field.
One of the earliest examples of a non-quasiprojective algebraic variety were given by Nagata.[3] Nagata's example was not complete (the analog of compactness), but soon afterwards he found an algebraic surface which was complete and non-projective.[4] Since then other examples have been found.