热带化多项式层
Tropicalizing polynomial strata
AI总结：
研究次数$D\geq2$复多项式分歧层${\rm Poly}_D(\vec\mu^*)$的热带化，通过构建带框装饰多项式树空间$\mathcal{T}_D(\vec\mu^*)$，利用多种方法解决热带化难题，证明其与相关骨架同构，实现紧致化并与其他紧致化比较。
AI中文摘要：
设${\rm Poly}_D(\vec\mu^*)$为次数$D\geq2$的复多项式参数空间中由具有分歧轮廓$\vec\mu^*$的多项式组成的分歧层。本文引入带框装饰多项式树的空间$\mathcal{T}_D(\vec\mu^*)$，并将其与${\rm Poly}_D(\vec\mu^*)$的动力学热带化等同起来。因该层没有先前已知的恰当环面紧致化，热带化无法先验得到。通过将${\rm Poly}_D(\vec\mu^*)$与刚性化带框赫维茨空间等同，利用扭曲容许覆盖紧致化及额外刚性化数据构建恰当环面紧致化及相关的贝科夫斯基骨架，证明$\mathcal{T}_D(\vec\mu^*)$与之同构，还表明射影化树空间$\mathbb P\mathcal{T}_D(\vec\mu^*)$紧致化了${\rm Poly}_D(\vec\mu^*)$，最后将此紧致化与多项式树射影化空间的德马尔科 - 麦克马伦紧致化进行比较。
英文摘要：
Let ${\rm Poly}_D(\vecμ^*)$ be the ramification stratum in the parameter space of degree $D \geq 2$ complex polynomials consisting of polynomials with ramification profile $\vecμ^*$. In this paper, we introduce a space $\mathcal{T}_D(\vecμ^*)$ of framed decorated polynomial trees and identify it with the dynamical tropicalization of ${\rm Poly}_D(\vecμ^*)$. A non-trivial point is that the tropicalization of ${\rm Poly}_D(\vecμ^*)$ is not available a priori, as the stratum does not come with a previously known proper toroidal compactification. We resolve this issue by identifying ${\rm Poly}_D(\vecμ^*)$ with a rigidified framed Hurwitz space. Using the twisted admissible cover compactification, together with additional rigidifying data, we construct a proper toroidal compactification and hence an associated Berkovich skeleton. We prove that $\mathcal{T}_D(\vecμ^*)$ is isomorphic to this skeleton. We further show that the projectivized tree space $\mathbb P\mathcal{T}_D(\vecμ^*)$ compactifies ${\rm Poly}_D(\vecμ^*)$. Finally, we compare this compactification with the DeMarco--McMullen compactification of polynomial moduli by projectivized space of polynomial trees.
