Stable n-pointed Curves

Abstract

Our main object of study will be the spaces $\overline M _{0,n} (\mathbb{P}^r ,d)$ of stable maps, whose introduction we defer to Section 2.3. Many of the properties of these spaces are inherited from $\overline M _{0,n}$, the important Deligne-Mumford-Knudsen moduli space of stable n-pointed rational curves which are the subject of this first chapter. We shall not go into the detail of the construction of $\overline M _{0,n}$, but content ourselves with the cases n≤5. The combinatorics of the boundary deserves a careful description. The principal reference for this chapter is Knudsen [51]; see also Keel [47].