We call a sheaf on an algebraic variety immaculate if it lacks any cohomology including the zero-th one, that is, if the derived version of the global section functor vanishes. Such sheaves are the basic tools when building exceptional sequences, investigating the diagonal property, or the toric Frobenius morphism. In the present paper we focus on line bundles on toric varieties. First, we present a possibility of understanding their cohomology in terms of their (generalized) momentum polytopes. Then we present a method to exhibit the entire locus of immaculate divisors within the class group. This will be applied to the cases of smooth toric varieties of Picard rank two and three and to those being given by splitting fans. The locus of immaculate line bundles contains several linear strata of varying dimensions. We introduce a notion of relative immaculacy with respect to certain contraction morphisms. This notion will be stronger than plain immaculacy and provides an explanation of some of these linear strata.

中文翻译：

---

环面品种上的完美线束

我们称代数簇上的一个层是完美无暇的，如果它缺乏任何上同调，包括第零个，也就是说，如果全局截面函子的导出版本消失了。在构建异常序列、研究对角线性质或复曲面 Frobenius 态射时，此类滑轮是基本工具。在本文中，我们关注复曲面品种的线束。首先，我们提出了根据它们的（广义）动量多胞体来理解它们的上同调的可能性。然后我们提出了一种方法来展示类组中完美除数的整个轨迹。这将适用于 Picard 等级 2 和 3 的平滑复曲面品种的情况以及由分裂粉丝给出的情况。完美线束的轨迹包含几个不同维度的线性地层。我们引入了关于某些收缩态射的相对完美的概念。这个概念将比简单的完美无瑕更强大，并提供了对这些线性地层中的一些的解释。