Every rationally connected variety over the function field of a curve has a rational point
Every rationally connected variety over the function field of a curve has a rational point
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曲线函数域上的每个有理连通簇都有一个有理点
DOI：
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发表时间：
2003
期刊：
影响因子：
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通讯作者：
Jason Starr
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文献类型：
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作者：
A. J. D. Jong;Jason Starr
在1992年的一篇论文中，Kollar，Miyaoka和Mori提出了以下问题：给定一个真平坦态射f：P → X，目标是一条非奇异曲线，其几何通属纤维是有理连通的，是否可以得出f允许一个正则截面？在基场是特征为零的代数闭场的情况下，格雷伯、哈里斯和斯塔尔用拓扑论证肯定地回答了这一问题。证明了当基域是任意特征的代数闭域时，f有正则截面。证明中的关键要素如下：（a）命题2.1表明，在取f与一般的代数态射π：Y → X的约化纤维积之后，我们可以假设f的光滑轨迹与每个几何纤维相交，以及（B）命题1.1是Graber-Harris-Starr中“移动分支点”论证的纯代数模拟。需要提醒读者的是，我们对“可分离理性联系”的定义与最初的定义略有不同。
In a paper from 1992, Kollar, Miyaoka and Mori posed the following question: Given a proper flat morphism f : P → X with target a nonsingular curve and whose geometric generic fiber is rationally-connected, does it follow that f admits a regular section? In the case that the ground field is an algebraically-closed field of characteristic zero, this was answered affirmatively by Graber, Harris, and Starr using a topological argument. We prove that f admits a regular section when the ground field is an algebraically-closed field of arbitrary characteristic. The key ingredients in the proof are the following: (a) Proposition 2.1 which shows that after taking the reduced fiber product of f with a generically etale morphism π: Y → X , we may assume that the smooth locus of f intersects every geometric fiber, and (b) Proposition 1.1 which is a purely algebraic analogue of the "moving branch points" argument in Graber-Harris-Starr. The reader is cautioned that our definition of "separably rationally connected" differs slightly from the original definition.