Studies in Moduli Theory and Birational Geometry

这个项目的研究领域在于代数几何，数学的分支致力于几何形状称为代数品种，由多项式方程定义。代数几何在理论物理学中有着重要的应用，物理学家将代数簇视为我们宇宙精细结构的一部分。这在第一个主题，模论中尤其如此。这个理论研究了一个显著的现象，其中所有相同类型的代数簇的集合通常表现为一个代数簇，称为模空间。因此，在代数几何学中，把“有机体”的能力看作其自身是一个“有机体”的隐喻，不仅是一种隐喻，而且是一个严格而非常有用的事实。在这个项目中研究的另一个主题是双有理几何，在这里侧重于解决奇点，在这个项目中应用到家庭的品种。 奇异性的解决是一个基本的程序，其中“坏”点的代数品种被删除，并取代“好”点。阿布拉莫维奇将继续研究问题，在双有理几何，特别是问题的函子半稳定减少。在这里，阿布拉莫维奇和合作者将建立在Hironaka的奇异性解决方法的基础上，以解决品种家族的奇异性。长期目标包括将这一努力扩展到其他几何类别和奇异叶理。此外，阿布拉莫维奇将继续研究模空间。主要焦点是K3曲面的模和算术，其中希望找到K3模空间是代数双曲的情况;对数模的可表示性，其中寻求Artin标准的模拟;寻求描述某些稳定曲面的显式模;以及完成对数退化公式的长期项目。该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的知识价值和更广泛的影响审查标准进行评估的支持。

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.