$\overline{M_{0, n}}$型的合理性问题
The rationality problem for forms of $\overline{M_{0, n}}$
摘要：
设$X$是定义在域$F$上的度为$5$的del Pezzo曲面。Yu的一个定理I. Manin和P. Swinnerton-Dyer断言每个$5$次的Del Pezzo曲面都是有理的。在本文中，我们推广这个结果如下。回想一下，域$F$上的次数为$5$的del Pezzo曲面正是亏格为$0$的稳定曲线的模空间$\overline{M_{0, 5}}$的扭曲$F$ -形式，其中有$5$标记点。设$n \geq 5$是一个整数，$F$是一个特征$\neq 2$的无限域。很容易看出，$\overline{M_{0, n}}$的每一个扭曲的$F$ -形式在$F$上都是单有理的。证明了（a）若$n$是奇数，则$\overline{M_{0, n}}$的每一个扭曲$F$ -形式在$F$上是有理数. (b)若$n$是偶数，则存在$\overline{M_{0, n}}$的域扩张$F/k$和扭曲$F$ -形式$X$，使得$X$不是$F$上的收缩有理数.
英文摘要：
Let $X$ be a del Pezzo surface of degree $5$ defined over a field $F$. A theorem of Yu. I. Manin and P. Swinnerton-Dyer asserts that every Del Pezzo surface of degree $5$ is rational. In this paper we generalize this result as follows. Recall that del Pezzo surfaces of degree $5$ over a field $F$ are precisely the twisted $F$-forms of the moduli space $\overline{M_{0, 5}}$ of stable curves of genus $0$ with $5$ marked points. Suppose $n \geq 5$ is an integer, and $F$ is an infinite field of characteristic $\neq 2$. It is easy to see that every twisted $F$-form of $\overline{M_{0, n}}$ is unirational over $F$. We show that (a) if $n$ is odd, then every twisted $F$-form of $\overline{M_{0, n}}$ is rational over $F$. (b) If $n$ is even, there exists a field extension $F/k$ and a twisted $F$-form $X$ of $\overline{M_{0, n}}$ such that $X$ is not retract rational over $F$.
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