曲线变形虫与Lyashko-Looijenga映射
Amoebas of curves and the Lyashko-Looijenga map
摘要：
对于复曲面$X$中的任意曲线$\mathcal{V}$，我们研究了从$\mathcal{V}$到其紧化变形虫$μ(\mathcal{V})$的矩映射$μ$的临界轨迹$S(\mathcal{V})$.我们证明了对于固定完全线性系中的曲线$\mathcal{V}$，除了某些实余维$1$壁之外，临界轨迹$S(\mathcal{V})$是光滑的。当$\mathcal{V}$和$S(\mathcal{V})$光滑时，我们研究了对$(\mathcal{V},S(\mathcal{V}))$的拓扑分类.作为主要工具，我们使用相对于对数高斯映射$γ: \mathcal{V} \rightarrow \mathbb{C}P^1$的Lyashko-Looijenga映射（$\mathcal{LL}$）。我们证明了两个声明有关$\mathcal{LL}$，这是至关重要的，我们的研究：地图$\mathcal{LL}$是代数的地图$\mathcal{LL}$延伸到节点曲线。它允许我们通过扰动节曲线来构造许多对$(\mathcal{V},S(\mathcal{V}))$的例子。
英文摘要：
For any curve $\mathcal{V}$ in a toric surface $X$, we study the critical locus $S(\mathcal{V})$ of the moment map $μ$ from $\mathcal{V}$ to its compactified amoeba $μ(\mathcal{V})$. We show that for curves $\mathcal{V}$ in a fixed complete linear system, the critical locus $S(\mathcal{V})$ is smooth apart from some real codimension $1$ walls. We then investigate the topological classification of pairs $(\mathcal{V},S(\mathcal{V}))$ when $\mathcal{V}$ and $S(\mathcal{V})$ are smooth. As a main tool, we use the Lyashko-Looijenga mapping ($\mathcal{LL}$) relative to the logarithmic Gauss map $γ: \mathcal{V} \rightarrow \mathbb{C}P^1$. We prove two statements concerning $\mathcal{LL}$ that are crucial for our study: the map $\mathcal{LL}$ is algebraic; the map $\mathcal{LL}$ extends to nodal curves. It allows us to construct many examples of pairs $(\mathcal{V},S(\mathcal{V}))$ by perturbing nodal curves.
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