Minimal Models of Moduli Spaces

DMS-0354994Sean M. KeelThe建议的研究中心在数学和理论物理学的基本对象的研究：超平面安排的模空间，曲线的模空间，和阿贝尔簇的模空间。本研究涉及森氏理论基本思想的应用。例如，提案的最大部分，以及具有最大潜在收益的部分与找到具有合理奇点的超平面排列的模的自然紧化（即，射影空间中超平面的有序n元组的模射影空间的模自同构）有关。这是有趣的，例如，因为明显的紧化-那些来自几何不变理论，有边界与字面上任意奇点（所有可能的仿射方案出现作为边界层），所以一个自然的去奇异化将包括一次所有奇点的去奇异化。研究者初步观察到超平面排列的模空间是（log）最小的，因此Mori理论的主要结论预言了一个具有合理边界的正则紧化的存在.主要的问题是找到这种紧化。研究人员建议考虑数学中一些基本参数化空间的几何--所谓的超平面安排的模空间、黎曼曲面和阿贝尔簇。该提案的统一主题是将几何学的一个分支--森理论--的基本思想应用于其他分支，这些分支的思想尚未得到广泛利用。最简单和潜在的最深刻的例子是在第一和主要部分的建议，研究人员观察到，森理论预测存在一个绝对规范的对象的解决奇异性-一个中心问题的几何。

DMS-0354994Sean M. KeelThe proposed research centers on the study of fundamental objects in mathematics and theoretic physics: moduli spaces of hyperplane arrangements, moduli spaces of curves, and moduli spaces of Abelian varieties. The research involves the application of basic ideas of Mori theory. For example the largest portion of the proposal, and the portion with the largest potential payoff has to do with finding a natural compactification of themoduli of hyperplane arrangements (i.e. moduli of ordered n-tuples of hyperplanes in projective space modulo automorphisms of the projective space) with reasonable singularities. This is of interest for example because the obvious compactifications -those that come from geometric invariant theory, have boundary with literally arbitrary singularities (all possible affine schemes occur as boundary strata), so a natural desingularisation would include a desingularisation of all singularities at once. The researcher makes the elementary observation that the moduli space of hyperplane arrangements is (log) minimal, and thus the main conjectures of Mori theory predict the existence of a canonical compactification, with reasonable boundary. The main question is to find this compactification.The researcher proposes to consider the geometry of some of the basic parameterising spaces in mathematics -- the so-called moduli spaces of hyperplane arrangements, Riemann surfaces, and Abelian varieties. The unifying theme of the proposal is the application of fundamental ideas from one branch of geometry, Mori theory, to other branches, where the ideas have not yet been widely exploited. The simplest and potentially deepest example is in the first and main part of the proposal, where the researcher observes that Mori theory predicts the existence of an absolutely canonical object for the resolution of singularities -- one of the central problems in geometry.