Birational Models of Singular Fano 3-folds

Fano簇在3维上的双有理分类，即所谓的3-folds，几十年来一直是数学中的一个具有挑战性的问题。两个簇被称为双有理的，如果他们可以确定后，删除一些小的（代数）子集。这个项目旨在阐明奇异Fano 3-folds的双有理几何。Fano簇，这里的研究对象，是基本的几何形状描述为代数方程（多项式）的解集，使它们的几何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，任何可以由有理函数参数化的几何形状都可以用Fano簇近似。无奇点的Fano 3-folds已被广泛研究。奇点是法诺3重线上的一个点，在这个点上相切的概念就没有意义了，就像冰淇淋蛋卷的锋利边缘一样。最小模型程序，双有理几何中的主要工具，表明Fano 3-folds可能携带温和的奇点，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Fano 3-folds家族，从中我们只能构建出几百个，但大多数其他家族仍然神秘地未构建。这种阻塞可以解决：大多数未构建的Fano 3折叠很可能不是实心的。一个Fano簇称为立体的，如果它不能对一束或一张低维Fano簇的网是双有理的。非固体Fano 3-folds因此不太有趣，因为铅笔模型提供了更多的几何信息。未构造的Fano 3-folds的代数结构类似于我们已知的那些，但具有强加的终端奇点：它们都有复反正则环。我们将研究奇异Fano 3-folds的实性，以便更好地理解数学的这个神秘角落。

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.