https://en.wikipedia.org/wiki/Glossary_of_algebraic_geometry#immersion Glossary of algebraic geometry
point A scheme S is a locally ringed space, so a fortiori a topological space, but the meanings of point of S are threefold:
a point P of the underlying topological space; a T -valued point of S is a morphism from T to S , for any scheme T ; a geometric point, where S is defined over (is equipped with a morphism to) Spec(K) , where K is a field, is a morphism from Spec ({￣K}) to S where {￣K} is an algebraic closure of K.
Geometric points are what in the most classical cases, for example algebraic varieties that are complex manifolds, would be the ordinary-sense points. The points P of the underlying space include analogues of the generic points (in the sense of Zariski, not that of Andre Weil), which specialise to ordinary-sense points. The T -valued points are thought of, via Yoneda's lemma, as a way of identifying S with the representable functor h_{S} it sets up. Historically there was a process by which projective geometry added more points (e.g. complex points, line at infinity) to simplify the geometry by refining the basic objects. The T -valued points were a massive further step. As part of the predominating Grothendieck approach, there are three corresponding notions of fiber of a morphism: the first being the simple inverse image of a point. The other two are formed by creating fiber products of two morphisms. For example, a geometric fiber of a morphism S^’ → S is thought of as
S^’ ×_{S} Spec({￣K}) .
This makes the extension from affine schemes, where it is just the tensor product of R-algebras, to all schemes of the fiber product operation a significant (if technically anodyne) result.
