## Osaka Journal of Mathematics

### 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 $\bar{M}_{0,n}$.

#### Article information

Source
Osaka J. Math., Volume 48, Number 4 (2011), 1115-1140.

Dates
First available in Project Euclid: 11 January 2012

https://projecteuclid.org/euclid.ojm/1326291221

Mathematical Reviews number (MathSciNet)
MR2871297

Zentralblatt MATH identifier
1253.14029

Subjects
Primary: 14H10: Families, moduli (algebraic)

#### Citation

Kiem, Young-Hoon; Moon, Han-Bom. Moduli spaces of weighted pointed stable rational curves via GIT. Osaka J. Math. 48 (2011), no. 4, 1115--1140. https://projecteuclid.org/euclid.ojm/1326291221

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