functorofpoints

The functor of points is a foundational perspective in algebraic geometry that encodes a scheme by describing its points over all test schemes. For a fixed base scheme S, the functor of points of a scheme X over S is the covariant functor X(-): Sch/S → Sets given by X(T) = Hom_S(T, X) for each T in Sch/S. When the base is a field k, this specializes to X(T) = Hom_k(T, X). In this language, a T-point of X is a morphism from T to X.

The viewpoint rests on the Yoneda lemma: a scheme X is determined by its functor of points, and conversely any functor that is representable by a scheme arises as the functor of points of that scheme. This viewpoint emphasizes families of points parameterized by varying base schemes rather than isolated geometric points.

A central motivation is the description of moduli problems. A T-point of X can be read as a family of geometric points of X parameterized by T, or, in more general settings, as a family of objects over T. This leads to moduli functors such as the Hilbert functor or the Picard functor, which assign to each T the set of subschemes or line bundles on T, respectively. If a moduli problem cannot be represented by a scheme, one extends the language to stacks, where one uses groupoid-valued functors to remember automorphisms of the objects.

Practically, the functor of points formalism is compatible with base change, fiber products, and universal properties, making it a powerful language for describing maps between schemes and for formulating and solving problems in families.

Examples and extensions include the functors of points for projective space and Grassmannians, as well as more general representable functors and their non-representable moduli counterparts.