for questions on one dimensional algebraic varieties over any field, including questions of moduli, and questions about specific curves.
An algebraic curve is a one dimensional algebraic variety. That is, it is a one dimensional subset of affine or projective space that is given by the solutions to a collection of polynomial equations. An algebraic curve over the complex numbers is a Riemann surface - more precisely, there is an analytification functor that yields finite type Riemann surfaces.
Basic results for algebraic curves include:
- the antiequivalence between proper curves and finitely generated extensions of the base field of transcendence degree 1.
- The Riemann-Roch theorem, which describes the space of rational functions with prescribed zeroes and poles.
- The Riemann-Hurwicz theorem, which describes ramification of maps between curves.
Algebraic curves have a single discrete invariant, called the genus. Curves of genus g are parametrized by a moduli space of dimension 3g-3.