论文标题
Voronoi砖块，曲折布置和线束的变性II
Voronoi tilings, toric arrangements and degenerations of line bundles II
论文作者
论文摘要
我们描述了与欧几里得空间的伏诺伊斜利相关的曲折排列方面的线束限制。这些整理编码有关可能无限多个限制之间关系的信息，并最终引起了极限线性序列的新定义。本文及其第一和第三部分伴侣部分是旨在探索这种新方法的系列中的第一个。 在第一部分中，我们设置了组合框架，并展示了与与边缘相关的整数长度加权的图形如何通过与图形本身及其子图相关的多面体来提供欧几里得空间的瓷砖。 在这一部分中，我们描述了与这些瓷砖相关的复曲面品种的布置。粗略地说，瓷砖中每个多层的正常风扇对应于感谢您的品种，这些感谢您的品种根据多面体的相遇方式将其粘合在一起。我们从不同的角度从不同的角度说明了这些曲折的布置：通过使用普通风扇作为圆环轨道的工会，通过描述（无限多个）多项式方程（在弹性线的双线无限链中定义它们），以及代数托里（Tori）的变性。 这些结果将在第三部分中使用，以实现我们描述沿着曲线衰败的系列束系的所有稳定限制。
We describe limits of line bundles on nodal curves in terms of toric arrangements associated to Voronoi tilings of Euclidean spaces. These tilings encode information on the relationship between the possibly infinitely many limits, and ultimately give rise to a new definition of limit linear series. This article and its first and third part companion parts are the first in a series aimed to explore this new approach. In the first part, we set up the combinatorial framework and showed how graphs weighted with integer lengths associated to the edges provide tilings of Euclidean spaces by polytopes associated to the graph itself and to its subgraphs. In this part, we describe the arrangements of toric varieties associated to these tilings. Roughly speaking, the normal fan to each polytope in the tiling corresponds to a toric variety, and these toric varieties are glued together in an arrangement according to how the polytopes meet. We provide a thorough description of these toric arrangements from different perspectives: by using normal fans, as unions of torus orbits, by describing the (infinitely many) polynomial equations defining them in products of doubly infinite chains of projective lines, and as degenerations of algebraic tori. These results will be of use in the third part to achieve our goal of describing all stable limits of a family of line bundles along a degenerating family of curves.
