Curves
Definition 29.41.1. Let f:X→S be a morphism of schemes.
We say f is proper if f is separated, finite type, and universally closed.
Definition 26.21.3. Let f:X→S be a morphism of schemes.
We say f is separated if the diagonal morphism ΔX/S is a closed immersion.
Definition 29.15.1. Let f:X→S be a morphism of schemes.
- We say that f is of finite type at x∈X if there exists an affine open neighbourhood Spec(A)=U⊂X of x and an affine open Spec(R)=V⊂S with f(U)⊂V such that the induced ring map R→A is of finite type.
- We say that f is locally of finite type if it is of finite type at every point of X.
- We say that f is of finite type if it is locally of finite type and quasi-compact.
Definition 26.20.1. A morphism of schemes f:X→S is said to be universally closed if every base change f′:XS′→S′ is closed.
Definition 33.3.1. Let k be a field. A variety is a scheme X over k such that X is integral and the structure morphism X→Spec(k) is separated and of finite type.
Lemma 33.43.10. Let X be a curve over k. Then either X is an affine scheme or X is H-projective over k.
Definition 44.6.3. Let k be a field. Let X be a smooth projective geometrically irreducible curve over k. The genus of X is g=dimkH1(X,X).