Birational Geometry and Topology of singular Fano 3-folds.

代数变体的分类问题一直是代数几何的一个基本问题。如果两个变种在去掉一个小子集之后变成同构的，那么它们就是双等价的。通过简单的双族运算（比如放大子类）就可以产生更大的变种，因此对变种进行分类就等于找到一个双族等价类的最佳或最小代表。最小模型程序（MMP）是20世纪70年代开始的一个尚未完成的项目，它给定一个代数变量X，执行有限个初等步骤来产生一个纯几何型的最终产品。这些纯几何类型一方面是最小模型，另一方面是范诺变种。最小模型，顾名思义，实现了与其等价类的最佳（最小）匹配的希望。范诺变异体接近于射影空间，应该被认为是黎曼曲面均匀化定理中球面的高维类比。假设存在MMP，那么品种分类问题就简化为理解MMP的基本步骤及其可能的结果。要完成更高维度的MMP，还有许多悬而未决的问题。在第三维度，MMP是在80年代完成的，但我们对其产品的了解最多是部分的。一些非常自然的问题仍未得到解答。例如，由于MMP的最终产品不是唯一的，那么MMP的两种可能的最终产品在什么情况下是双重的？是否有可能分辨出哪些最终产品是理性的，即与投影空间的关系？我的研究旨在为Fano 3-fold回答这些问题。MMP产生轻度奇异的变种——在维数3中，这些奇异点是孤立的点。我的研究表明，当一个Fano有“许多”奇异点时，它倾向于获得许多到其他Fano 3-fold的两国映射，因此表现得像射影空间。在这种情况下，“许多”的意思是拓扑的：一个Fano有许多奇点，如果这些奇点实际上位于包含在X中的曲面S上，而这个曲面S不是X的超平面部分。我的研究项目认为，反过来说，如果X上没有这样的曲面，那么X的行为就好像是非奇异的。令人惊讶的是，对于小程度的法诺，这通常意味着它们只与很少的其他法诺3-fold有血缘关系，因此是非理性的。

The classification of algebraic varieties up to birational equivalence has long been a fundamental problem of Algebraic Geometry.Two varieties are birationally equivalent if they become isomorphic after removing a small subset. It is possible to produce ever larger varieties by simple birational operations (such as blowing up subvarieties), and hence classifying varieties amounts to finding a best, or ``minimal , representative for a birational equivalence class. The Minimal Model Program (MMP) is a still incomplete project started in the 1970s, which given an algebraic variety X, performs a finite number of elementary steps to produce an end product of pure geometric type. These pure geometric type are minimal models on the one hand, and Fano varieties on the other.Minimal models, as their name indicates, realise the hope of being a best (minimal) match for their equivalence class. Fano varieties are close to projective spaces, and should be thought of as the higher dimensional analogue of the sphere in the Uniformisation theorem for Riemann surfaces. Assuming the MMP, the problem of classification of varieties is reduced to understanding the elementary steps of the MMP and its possible outcomes. There remain a number of open questions to achieve completion of the MMP in higher dimensions. In dimension 3, the MMP was completed in the 80s, yet our understanding of its products is partial at best. Some very natural questions remain unanswered. For instance, since the end product of the MMP is not unique, when are two possible end products of the MMP birational? Is it possible to tell which end products are rational, i.e. birational to projective space? My research aims at answering these questions for Fano 3-folds. The MMP produces varieties that are mildly singular-- in dimension 3, these singularities are isolated points. My research shows that when a Fano has ``many'' singular points it tends to acquire many birational maps to other Fano 3-folds, and therefore behave like projective space. What ``many'' means in this context is topological: a Fano has many singular points if these singular points actually lie on a surface S contained in X that is not a hyperplane section of X. My research project argues that, conversally, if there is no such surface lying on X, X behaves as if it was nonsingular. Surprisingly, for Fanos of small degree, this often implies that they are only birational to very few other Fano 3-folds and are therefore nonrational.