Functor represented by a scheme

In algebraic geometry, a functor represented by a scheme X is a set-valued contravariant functor on the category of schemes such that the value of the functor at each scheme S is (up to natural bijections, or one-to-one correspondence) the set of all morphisms S → X {\displaystyle S\to X} . The functor F is then said to be naturally equivalent to the functor of points of X; and the scheme X is said to represent the functor F, and to classify geometric objects over S given by F. A functor producing certain geometric objects over S might be represented by a scheme X. For example, the functor taking S to the set of all line bundles over S (or more precisely n-dimensional linear systems) is represented by the projective space X = P n − 1 {\displaystyle X=\mathbb {P} ^{n-1}} .

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