(Marked) Genus-zero curves

A smooth connected curve of arithmetic genus zero is isomorphic to P1. A connected curve of arithmetic genus zero is isomorphic to a collection of P1s, joined at simple nodes, whose dual graph is a tree.

An n-marked genus-zero curve is a connected genus-zero curve, together with the data of n distinct smooth points p1,...,pn.

Properties

 * The automorphism group of P1 is isomorphic to PGL(2). Given three distinct (ordered) points of P1, there is a unique element of Aut(P1) sending those points to 0, 1, and \infty.

Related

 * Genus-zero stacky curves

Stable marked genus-zero curves
A connected marked curve of arithmetic genus zero is stable if every component of the normalization contains at least three points that are either nodes or marked points. Alternatively, if the dual tree is enriched with half-edges corresponding to marked points, then every vertex must have valence at least 3.

Moduli
The moduli stack M0n of n-marked smooth genus zero curves is a smooth (n-3)-dimensional stack. It is Deligne-Mumford if and only if n≥3; in this case it is a smooth quasiprojective variety, isomorphic to (C\{0,1,\infty})n-3 with all diagonals removed.

The moduli stack Mbar0n of n-marked stable genus-zero curves is a smooth irreducible projective compactification of M0n for n≥3.