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. 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