arXiv 1703.030672017-03math.AG

代数曲线的阿贝尔覆盖的热带化

Tropicalizing abelian covers of algebraic curves

Paul Alexander Helminck

备注：: 2nd version: PhD thesis version

摘要：

本文研究了离散域$K$上代数曲线的Berkovich骨架。我们使用投影线的覆盖$C\rightarrow{\mathbb{P}^{1}}$来实现这一点。为了研究这些覆盖，我们取函数域$K(\mathbb{P}^{1})\rightarrow{K(C)}$的相应注入的伽罗瓦闭包，给出了一个伽罗瓦态射$\overline{C}\rightarrow{\mathbb{P}^{1}}$ . Liu和Lorenzini的一个定理告诉我们如何将半稳定模型$\mathcal{C}\rightarrow{\mathcal{D}}$的伽罗瓦态射与这个态射联系起来。也就是说，我们在$\mathcal{D}$的特殊纤维中使分支轨迹不相交，并去除$\mathcal{D}_{s}$的组件上的任何垂直分支。这个态射$\mathcal{C}\rightarrow{\mathcal{D}}$然后产生交集图的态射$Σ(\mathcal{C})\rightarrow{Σ(\mathcal{D})}$。我们的目标是从$Σ(\mathcal{D})$重建$Σ(\mathcal{C})$，我们将通过给出一组覆盖和扭曲数据来实现这一点。这些然后给算法找到的Berkovich骨架的曲线$C$时，曲线有一个态射$\overline{C}\rightarrow{\mathbb{P}^{1}}$与一个可解的伽罗瓦群。特别地，给出了求任意亏格三曲线的Berkovich骨架的算法。这些覆盖也给出了一个经典结果的一个新的证明半稳定约化型的椭圆曲线，说，一个椭圆曲线有潜在的良好的减少，当且仅当$j$ -不变量的价值是积极的。

英文摘要：

In this thesis, we study the Berkovich skeleton of an algebraic curve over a discretely valued field $K$. We do this using coverings $C\rightarrow{\mathbb{P}^{1}}$ of the projective line. To study these coverings, we take the Galois closure of the corresponding injection of function fields $K(\mathbb{P}^{1})\rightarrow{K(C)}$, giving a Galois morphism $\overline{C}\rightarrow{\mathbb{P}^{1}}$. A theorem by Liu and Lorenzini tells us how to associate to this morphism a Galois morphism of semistable models $\mathcal{C}\rightarrow{\mathcal{D}}$. That is, we make the branch locus disjoint in the special fiber of $\mathcal{D}$ and remove any vertical ramification on the components of $\mathcal{D}_{s}$. This morphism $\mathcal{C}\rightarrow{\mathcal{D}}$ then gives rise to a morphism of intersection graphs $Σ(\mathcal{C})\rightarrow{Σ(\mathcal{D})}$. Our goal is to reconstruct $Σ(\mathcal{C})$ from $Σ(\mathcal{D})$ and we will do this by giving a set of covering and twisting data. These then give algorithms for finding the Berkovich skeleton of a curve $C$ whenever that curve has a morphism $\overline{C}\rightarrow{\mathbb{P}^{1}}$ with a solvable Galois group. In particular, this gives an algorithm for finding the Berkovich skeleton of any genus three curve. These coverings also give a new proof of a classical result on the semistable reduction type of an elliptic curve, saying that an elliptic curve has potential good reduction if and only if the valuation of the $j$-invariant is positive.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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