Definition
Algebraic curve
A separated finite-type scheme of pure dimension one over a field.
Definition
In this knowl, an algebraic curve over a field is a separated scheme over of finite type and pure dimension . Some authors additionally require a curve to be reduced, irreducible, or geometrically integral, so those hypotheses must be stated when they matter.
Common refinements
A curve may be affine or proper, smooth or singular, connected or disconnected. A smooth projective curve is the standard base in the unramified geometric Langlands program.
When , the analytification of a smooth proper connected algebraic curve is a compact Riemann surface. Removing finitely many closed points produces a punctured curve, the usual base for ramified local systems.
References
- The Stacks Project Authors, “Curves,” chapter reference.