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Birational Geometry and Topology of singular Fano 3-folds.

奇异 Fano 3 倍的双有理几何和拓扑。

基本信息

批准号：
EP/H028811/2
负责人：
Anne-Sophie Kaloghiros
金额：
$ 22.17万
依托单位：
Imperial College London
依托单位国家：
英国
项目类别：
Fellowship
财政年份：
2012
资助国家：
英国
起止时间：
2012 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FH028811%2F2
关键词：
Birational Geometry Topology singular Fano

项目摘要

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.

代数簇的分类直至双有理等价长期以来一直是代数几何的一个基本问题。如果两个簇在去除一个小子集后变得同构，则它们是双有理等价的。通过简单的双有理操作（例如放大子品种）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寻找最佳或“最小”代表。最小模型程序 (MMP) 是一个始于 20 世纪 70 年代的尚未完成的项目，它给定代数簇 X，执行有限数量的基本步骤以产生纯几何类型的最终产品。这些纯几何类型一方面是最小模型，另一方面是 Fano 变体。最小模型，正如其名称所示，实现了成为其等价类的最佳（最小）匹配的希望。法诺簇接近射影空间，应被视为黎曼曲面均匀化定理中球体的高维类似物。假设采用 MMP，品种分类问题就简化为了解 MMP 的基本步骤及其可能的结果。为了完成更高维度的 MMP，仍有许多悬而未决的问题。在维度3中，MMP是在80年代完成的，但我们对其产品的了解充其量是片面的。一些非常自然的问题仍未得到解答。例如，由于 MMP 的最终产物不是唯一的，那么 MMP 的两个可能的最终产物何时是双有理的？是否有可能判断哪些最终产品是有理的，即射影空间的双有理？我的研究旨在回答 Fano 3 倍的这些问题。 MMP 产生的变量具有轻微的奇异性——在维度 3 中，这些奇异性是孤立的点。我的研究表明，当 Fano 具有“许多”奇点时，它往往会获得许多其他 Fano 3 重的双有理映射，因此表现得像射影空间。在这种情况下，“许多”意味着拓扑：Fano 具有许多奇点，如果这些奇点实际上位于 X 中包含的曲面 S 上，而该曲面 S 不是 X 的超平面部分。我的研究项目认为，相反，如果 X 上不存在这样的曲面，则 X 的行为就好像它是非奇异的。令人惊讶的是，对于小阶 Fanos，这通常意味着它们仅对极少数其他 Fano 3 倍是双有理的，因此是非有理的。