Towards Mori’s program for the moduli space of stable maps

We introduce and compute the class of a number of effective divisors on the moduli space of stable maps $\overline{{\scr M}}_{0,0}({\Bbb P}^{r}, d)$, which, for small $d$, provide a good understanding of the extremal rays and the stable base locus decomposition for the effective cone. We also discuss various birational models that arise in Mori's program, including the Hilbert scheme, the Chow variety, the space of $k$-stable maps, the space of branchcurves and the space of semi-stable sheaves.