论文标题
野生情况下的簇曲线和半纤维化曲线模型
Clusters and semistable models of hyperelliptic curves in the wild case
论文作者
论文摘要
给定在完整的离散估值字段$ k $的Galois封面$ y \至x $的x $，带有代数封闭的残基字段，我们定义了$ y $ y $ y $ $ y $的$ y $的$ y $的$ y $ y $ y $ y $ y $ y $ y $ y $ \ y} $ \ mathcal} $ {y}^y} $ y $，我们讨论其属性。当$ y：y^2 = f（x）$是一条过度纤维化曲线时，我们将重点放在案例上，被视为photchive Line $ x $ x的$ 2 $封面$ x：= \ mathbb {p} _k^1 $，并展示了一种计算相对稳定模型的实用方法。对于残留特征$ p \ neq 2 $，特殊光纤$（\ nathcal {y}^{\ sathrm {\ mathrm {rst}}）_ s $恰恰相对于非固定型$ f $ $ f $ $ f $ f ynoth of $ f y ynother的其他$ f $ f的非辛格尔顿根类别恰恰相对应。关于分裂场的诱导离散估值；但是，这种关系在$ P = 2 $案例中的简单明了，这是我们的主要重点（尽管如此，我们引入的技术也使我们能够在$ p \ neq 2 $ case中恢复更简单，已知的结果）。我们表明，当$ p = 2 $时，对于包含$ f $的均匀数的每个集群，有$ 0 $，$ 1 $或$ 2 $的组件的$（\ Mathcal {y}^{\ Mathrm {rst}}）_ s $与之相关，我们确定与之相关的方法。我们还定义了k [t] $中的多项式$ f（t）\，其根源使我们能够找到$（\ mathcal {y}^{\ mathrm {rst}}）_ s $的组件，这些组件未连接到偶性簇。
Given a Galois cover $Y \to X$ of smooth projective geometrically connected curves over a complete discrete valuation field $K$ with algebraically closed residue field, we define a semistable model of $Y$ over the ring of integers of a finite extension of $K$, which we call the relatively stable model $\mathcal{Y}^{\mathrm{rst}}$ of $Y$, and we discuss its properties. We focus on the case when $Y : y^2 = f(x)$ is a hyperelliptic curve, viewed as a degree-$2$ cover of the projective line $X := \mathbb{P}_K^1$, and demonstrate a practical way to compute the relatively stable model. In the case of residue characteristic $p \neq 2$, the components of the special fiber $(\mathcal{Y}^{\mathrm{rst}})_s$ correspond precisely to the non-singleton clusters of roots of the defining polynomial $f$, i.e. the subsets of roots of $f$ which are closer to each other than to the other roots of $f$ with respect to the induced discrete valuation on the splitting field; this relationship, however, is far less straightforward in the $p=2$ case, which is our main focus (the techniques we introduce nevertheless also allow us to recover the simpler, already-known results in the $p\neq 2$ case). We show that, when $p = 2$, for each cluster containing an even number of roots of $f$, there are $0$, $1$, or $2$ components of $(\mathcal{Y}^{\mathrm{rst}})_s$ corresponding to it, and we determine a direct method of finding and describing them. We also define a polynomial $F(T) \in K[T]$ whose roots allow us to find the components of $(\mathcal{Y}^{\mathrm{rst}})_s$ which are not connected to even-cardinality clusters.
