  We compare the Kontsevich moduli space of genus 0 stable maps to projective
space with the quasi-map space when $d=3$. More precisely, we prove that when
$d=3$, the obvious birational map from the quasi-map space to the moduli space
of stable maps is the composition of three blow-ups followed by two blow-downs.
Furthermore, we identify the blow-up/down centers explicitly in terms of the
moduli spaces for lower degrees. Using this, we calculate the Betti numbers,
the integral Picard group, and the rational cohomology ring. The degree two
case is worked out as a warm-up.
