Minimal Models of Moduli Spaces

建议的研究集中在数学和理论物理的基本对象的研究：超平面排列的模空间，曲线的模空间和阿贝尔变的模空间。该研究涉及到Mori理论基本思想的应用。例如，该建议的最大部分，以及具有最大潜在回报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。这是一个有趣的例子，因为明显的紧化——那些来自几何不变理论的紧化，具有具有任意奇点的边界（所有可能的仿射方案都作为边界层出现），因此自然的去具体化将包括一次所有奇点的去具体化。研究人员初步观察到超平面排列的模空间是（对数）极小的，因此Mori理论的主要猜想预测了正则紧化的存在，并且具有合理的边界。主要的问题是找到这个紧化。研究人员建议考虑数学中一些基本参数化空间的几何——所谓的超平面排列、黎曼曲面和阿贝尔变分的模空间。该提案的统一主题是将Mori理论这一几何学分支的基本思想应用于尚未被广泛利用的其他分支。最简单也可能是最深刻的例子是在提案的第一部分和主要部分，研究人员观察到Mori理论预测了解决奇点的绝对规范对象的存在——奇点是几何中的中心问题之一。

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.