Birational Geometry and Topology of singular Fano 3-folds.

代数簇的分类一直以来都是代数几何的一个基本问题，如果两个簇在去掉一个小的子集后变得同构，则它们在子族上是等价的。通过简单的混交操作(如炸毁亚种)就有可能生产出更大的品种，因此，对品种进行分类相当于为一个混血等值类别找到一个最好的或最小的代表性。最小模型程序是始于20世纪70年代的一个尚未完成的项目，它给定一个代数变量X，执行有限个基本步骤来产生纯几何类型的最终产品。这些纯几何类型一方面是极小模型，另一方面是Fano变体。正如它们的名字所示，极小模型实现了成为其等价类的最佳(最小)匹配的希望。Fano簇接近于射影空间，应该被认为是黎曼曲面的均匀化定理中球面的高维类比。假设有基质金属氧化物，品种分类的问题归结为理解基质金属氧化物酶的基本步骤及其可能的结果。要在更高的维度上完成基质金属氧化物酶，仍有许多悬而未决的问题。在维度3，基质金属氧化物是在80年代完成的，但我们对其产品的理解充其量也是片面的。一些非常自然的问题仍然没有得到回答。例如，由于基质金属氧化物的最终产物不是唯一的，那么基质金属氧化物的两种可能的最终产物是什么时候？是否有可能区分出哪些最终产品是有理的，即从二元空间到射影空间？我的研究旨在为Fano 3-Folds回答这些问题。在3维空间中，这些奇点是孤立的点。我的研究表明，当一个Fano有“许多”奇点时，它往往会获得许多到其他Fano 3-折叠的双射映射，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X中包含的S所包含的曲面上，而该曲面不是X的超平面截面。我的研究项目认为，反过来，如果X上没有这样的曲面，X的行为就像它是非奇异的一样。令人惊讶的是，对于小程度的FANO，这通常意味着它们只对极少数其他FANO三折是双数的，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.