Abstract curves

An abstract curve is a 1-dimensional reduced (sometimes irreducible) scheme (of finite type over C).

Properties

 * A smooth irreducible abstract curve can be viewed as a Riemann surface.
 * A connected abstract curve has an arithmetic genus (genus of a smoothing) and a geometric genus (genus of the normalization), which coincide when the curve is smooth. (They also coincide with the genus of the Riemann surface.)
 * A disconnected abstract curve also has arithmetic and geometric genera, which can be defined by adding the genera of components, and subtracting 1 for each component after the first.

Moduli

 * For any genus g, there is a moduli stack of smooth connected abstract curves of genus g. For g≥2, this stack is Deligne-Mumford and has quasi-projective coarse moduli space. The moduli stack is smooth and irreducible. It has a smooth compactification ...

Special cases

 * Genus-zero curves
 * Genus-one curves
 * Genus-two curves
 * Genus-three curves
 * Nodal curves
 * Marked curves
 * Stable (marked) curves