Curves over a field

Let $k$ be a field. A curve over $k$ is a $k$-scheme $C$ such that

1. $C$ is geometrically integral: the algebraic-closure fibre $C_{\bar{k}}=C{\times }_{\mathrm{Spec}k}\mathrm{Spec}\bar{k}$ of Geometric fibres and geometric points is integral, that is, reduced, irreducible and nonempty, in the sense of Geometric properties of fibres and Integral schemes;
2. $C$ is separated over $k$ (Separated morphism of schemes);
3. $C$ is of finite type over $k$ (Locally finite type and finite type morphisms);
4. the underlying topological space of $C$ has chain dimension one (Chain dimension and the empty-space convention).

A smooth curve is a curve $C$ whose structure morphism $C\to \mathrm{Spec}k$ is smooth (Smooth morphisms via local standard smooth presentations); a proper curve is a curve whose structure morphism is proper (Proper morphisms). Smoothness and properness are extra adjectives attached to a curve; neither is part of the meaning of the word curve, and a curve need be neither smooth nor proper.

Chain dimension one means that the underlying space admits a strict chain $Z_0⊊Z_1$ of nonempty irreducible closed subsets and admits no strict chain of length two; equivalently the space has Krull dimension $1$ and is not the empty space. The empty scheme is therefore not a curve.

The scheme $C$ is a $k$-scheme of dimension one in the sense that its irreducible components have dimension one. A curve is often written with its field of definition omitted when no confusion arises, and a smooth proper curve always means a curve that is both smooth and proper; the running convention of this page is that a claim about curve local rings, divisors, or genera names its hypotheses explicitly rather than hiding them in the word curve.