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arXiv 1809.07293math.AGmath.NT
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射影曲线的单值截群

Sectional monodromy groups of projective curves

Borys Kadets

备注：
to appear in Journal of the London Mathematical Society
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摘要：

在代数闭域$K$上固定一条度$d$投影曲线$X \subset \mathbb{P}^r$。设$U \subset (\mathbb{P}^r)^*$是稠密开子簇，使得每个超平面$H \in U$与$X$相交于$d$光滑点.改变$H \in U$产生单值作用$φ: π_1^{\text{ét}}(U) \to S_d$。设$G_X := \mathrm{im}(φ)$。置换群$G_X$称为$X$的截单值群。在特征零中，$G_X$总是全对称群，但在特征$p$中，分段单值群可以更小。对于一个大类的空间曲线（$r \geqslant 3$），我们分类的所有可能性的部分monodromy组$G$以及曲线与$G_X=G$。我们应用类似的方法来研究$\mathbb{P}^2$中的一个特殊的有理曲线族，这使我们能够回答一个关于一般三项式的伽罗瓦群的老问题。

英文摘要：

Fix a degree $d$ projective curve $X \subset \mathbb{P}^r$ over an algebraically closed field $K$. Let $U \subset (\mathbb{P}^r)^*$ be a dense open subvariety such that every hyperplane $H \in U$ intersects $X$ in $d$ smooth points. Varying $H \in U$ produces the monodromy action $φ: π_1^{\text{ét}}(U) \to S_d$. Let $G_X := \mathrm{im}(φ)$. The permutation group $G_X$ is called the sectional monodromy group of $X$. In characteristic zero $G_X$ is always the full symmetric group, but sectional monodromy groups in characteristic $p$ can be smaller. For a large class of space curves ($r \geqslant 3$) we classify all possibilities for the sectional monodromy group $G$ as well as the curves with $G_X=G$. We apply similar methods to study a particular family of rational curves in $\mathbb{P}^2$, which enables us to answer an old question about Galois groups of generic trinomials.

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