Birational Geometry and Topology of singular Fano 3-folds.

代数簇的分类一直到双有理等价一直是代数几何的基本问题，两个代数簇在去掉一个小子集后同构，则它们是双有理等价的。通过简单的双有理运算（如炸毁子品种）可以产生更大的品种，因此对品种进行分类相当于为双有理等价类找到一个最佳或最小的代表。最小模型程序（Minimal Model Program，MMP）是一个开始于20世纪70年代的尚未完成的项目，它给出一个代数簇X，执行有限数量的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变种。极小模型，正如它们的名字所示，实现了成为它们等价类的最佳（最小）匹配的希望。法诺簇接近于射影空间，并且应该被认为是黎曼曲面的统一化定理中的球面的高维类似物。假设MMP，品种分类的问题是减少到了解MMP的基本步骤和可能的结果。要在更高层面上完成小额供资和微型企业方案，仍有一些悬而未决的问题。在维度3中，MMP在80年代完成，但我们对其产品的理解充其量是部分的。一些非常自然的问题仍然没有答案。例如，由于MMP的最终产物不是唯一的，什么时候MMP的两个可能的最终产物是双有理的？是否有可能分辨出哪些最终产品是有理的，即对射影空间是双有理的？我的研究旨在回答这些问题的法诺3倍。MMP产生的变种是轻度奇异的-在3维，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，它倾向于获得许多到其他Fano 3-folds的双有理映射，因此表现得像射影空间。在这里，“许多”的意思是拓扑的：一个法诺有许多奇点，如果这些奇点实际上位于包含在X中的曲面S上，而曲面S不是X的超平面截面。我的研究项目认为，反过来说，如果X上没有这样的曲面，X的行为就好像它是非奇异的。令人惊讶的是，对于小度的Fanos，这通常意味着它们只对很少的其他Fano 3-folds是双有理的，因此是非有理的。

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.