Moduli spaces of weighted pointed stable rational curves via GIT

Abstract

We construct moduli spaces of weighted pointed stable rational curves $\bar{M}_{0,n \cdot \epsilon}$ with symmetric weight data by the GIT quotient of moduli spaces of weighted pointed stable maps $\bar{M}_{0,n \cdot \epsilon}(\mathbb{P}^{1},1)$. As a consequence, we prove that the Knudsen--Mumford space $\bar{M}_{0,n}$ of $n$-pointed stable rational curves is obtained by a sequence of explicit blow-ups from the GIT quotient $(\mathbb{P}^{1})^{n}\qquotient \mathit{SL}(2)$ with respect to the symmetric linearization $\mathcal{O}(1, \ldots, 1)$. The intermediate blown-up spaces turn out to be $\bar{M}_{0,n \cdot \epsilon}$ for suitable ranges of $\epsilon$. As an application, we provide a new unconditional proof of M. Simpson's theorem about the log canonical models of $\bar{M}_{0,n}$.