Studies in Moduli Theory and Birational Geometry

这个项目的研究领域是代数几何，这是一门专门研究几何形状的数学分支，称为代数簇，由多项式方程定义。代数几何在理论物理中有重要的应用，物理学家认为代数簇是我们宇宙精细结构的一部分。对于第一个主题，模理论尤其如此。这一理论研究了一个值得注意的现象，即所有相同类型的代数族的集合往往表现为一个代数族，称为模空间。因此，在代数几何中，把“有机体”的群体看作“有机体”本身就是一个“有机体”的比喻，不仅是一种非比喻，而且是一个严谨而相当有用的事实。这个项目中研究的另一个主题是二次几何，这里的重点是奇点的分解，在这个项目中应用于变种族。奇点分解是代数簇的“坏”点被“好”点取代的基本过程。阿布拉莫维奇将继续研究双曲几何中的问题，特别是函数半稳定约化问题。在这里，阿布拉莫维奇和合作者将建立在平中的奇点分解方法的基础上，以解决变种家族的奇点。长期目标包括将这一努力扩展到其他几何类别和单叶。此外，阿布拉莫维奇还将继续研究模空间。主要焦点是K3曲面的模和算术，其中人们希望找到K3模空间在代数上是双曲的情况；对数模的可表现性，其中寻求类似Artin的标准；寻求描述某些稳定表面的显式模；以及完成关于对数退化公式的长期项目。-该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The area of study of this project lies within algebraic geometry, thebranch of mathematics devoted to geometric shapes called algebraicvarieties, defined by polynomial equations. Algebraic geometryhas significant applications in theoretical physics, where physicists consideralgebraic varieties as a piece of the fine structure of ouruniverse. This is especially true with the first topic, modulitheory. This theory studies a remarkable phenomenon in which thecollection of all algebraic varieties of the same type is oftenmanifested as an algebraic variety, called a moduli space, in its ownright. Thus in algebraic geometry, the metaphor of thinking about acommunity of "organisms" as itself being an "organism" is not just ametaphor but a rigorous and quite useful fact. The other topic studied inthis project is birational geometry, focusing here on resolution of singularities, applied in this project to families of varieties. Resolution of singularities is a fundamental procedure where "bad" points of an algebraic variety are removed and replaced by "good" points.Abramovich will continue studying problems in birational geometry, specifically the problem of functorial semistable reduction. Here Abramovich and collaborators will build on Hironaka's method of resolution of singularities in order to resolve singularities of families of varieties. Long term goals include extending this effort to other geometric categories and to singular foliations. In addition, Abramovich will continue to study moduli spaces. The main foci are Moduli and arithmetic of K3 surfaces, where one wishes to find situations where K3 moduli spaces are algebraically hyperbolic; representability of logarithmic moduli, where an analogue of Artin's criteria is sought; a quest to describe explicit moduli of certain stable surfaces; and completion of a long-term project on the logarithmic degeneration formula.--------This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.