Abstract algebraic variety

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In algebraic geometry, an abstract algebraic variety is an algebraic variety defined intrinsically: that is, without reference to an external geometric context. Hartshorne (1976) defines an abstract variety to be an integral, separated scheme of finite type over an algebraically closed field k. One further defines an abstract projective variety to be an abstract variety which is a projective scheme over k, and an abstract quasiprojective variety to be an open subscheme of an abstract projective variety. Then, identifying classical algebraic varieties as schemes, in fact they are identified precisely with the abstract quasiprojective varieties.