Birational Models of Singular Fano 3-folds

三维Fano变量的两国分类，即所谓的三折，几十年来一直是数学中的一个具有挑战性的问题。如果两个变量可以在去掉一些小的（代数）子集后被识别，那么它们就被称为双变量。该项目旨在阐明奇异法诺三折的双几何。法诺变异体，这里的研究对象，是描述为代数方程（多项式）的解集的基本几何形状，因此它们的几何形状具有一些特殊的正性质。粗略地说，它们是正弯曲的。它们出现在应用中：例如，任何可以用有理函数参数化的几何形状都可以用法诺变量来近似。无奇点的Fano 3-fold已被广泛研究。奇点是法诺三折上的一个点，在这个点上切线的概念就没有意义了，就像冰淇淋蛋筒的锋利边缘。最小模型程序是双空间几何的主要工具，它表明法诺三折可能携带轻微的奇点，即所谓的终端奇点。因此，奇异模型的研究是至关重要的。我们知道，最多有52,000个法诺三叠系，从中我们只能构造出几百个，但其他大多数仍神秘地未被构造出来。这种阻塞是可以解决的：大多数未构建的Fano 3-fold很可能不是实心的。如果一个法诺变体不能与低维法诺变体的铅笔或网相对应，则称为固体。非实心Fano 3-fold因此不那么有趣，因为铅笔模型有更多的几何信息可以提供。未构造的Fano 3-fold的代数结构与我们所知道的相似，但附加了终端奇点：它们都具有复杂的复数反正则环。为了更好地理解这个数学的神秘角落，我们将研究奇异法诺三折的固体性。

The birational classification of Fano varieties in dimension 3, so-called 3-folds, has been a challenging problem in mathematics for decades. Two varieties are called birational if they can be identified after removing some small (algebraic) subsets from them. This project aims to shed light on the birational geometry of singular Fano 3-folds. Fano varieties, the objects of study here, are fundamental geometric shapes described as the solution sets of algebraic equations (polynomials) so that their geometry has some special positivity properties. Roughly speaking, they are positively curved. They appear in applications: for example, any geometric shape that can be parametrized by rational functions is approximated by Fano varieties. Fano 3-folds without singularities have been studied extensively. A singularity is a point on a Fano 3-fold at which the concept of tangency fails to make sense, like the sharp edge of an ice-cream cone. The Minimal Model Program, the main tool in birational geometry, indicates that Fano 3-folds may carry mild singularities, the so-called terminal singularities. Hence the study of singular models is vital. We know that there are at most 52,000 families of Fano 3-folds, from which we can construct only a few hundred, but most others remain mysteriously unconstructed. This obstruction may be resolved: most unconstructed Fano 3-folds are most likely not solid. A Fano variety is called solid if it cannot be birational to a pencil, or web, of lower dimensional Fano varieties. Non-solid Fano 3-folds are hence less interesting, as the pencil model has more geometric information to offer. The algebraic structure of the unconstructed Fano 3-folds is similar to those that we know but with imposed terminal singularities: they all have complex pluri anticanonical rings. We will examine solidity for the singular Fano 3-folds in order to develop a better understanding of this mysterious corner of mathematics.