We construct the Mumford-Knudsen space of n pointed stable rational curves by
a sequence of explicit blow-ups from the GIT quotient (P^1)^n//SL(2) with
respect to the symmetric linearization O(1,...,1). The intermediate blown-up
spaces turn out to be the moduli spaces of weighted pointed stable curves for
suitable ranges of weights. As an application, we provide a new unconditional
proof of M. Simpson's Theorem about the log canonical models of the
Mumford-Knudsen space. We also give a basis of the Picard group of the moduli
spaces of weighted pointed stable curves.Comment: An error (Lemma 5.3 in v1) has been corrected
