Birational Geometry of Moduli Spaces and Applications

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。主要研究者对模空间的几何研究感兴趣，特别是那些正交或酉型的模变体（例子包括K3曲面或低格曲线的模空间）。Laza的工作利用了模空间的多个双空间模型的存在性（例如，通过使用不同的紧化方法，如几何不变理论（GIT）或Hodge理论）来提取给定模空间的有用几何信息。首席研究员计划将这种想法应用到涉及模空间的各种项目中。PI提出要研究的第一个问题是找到极化K3曲面模的几何紧化问题。GIT商的变化方法和来自最小模型程序的思想为解决低次情况提供了一种有希望的方法。第二个项目解决了关于紧致超凯勒流形（K3曲面的高维类似物）模量的各种问题。特别地，PI计划从算术和几何的角度研究双EPW性的模空间（由O'Grady引入）的情况。第三个课题是关于四格曲线和三次三折的模空间的两族几何的一些具体问题。该建议的一般领域是代数几何，这是数学的一个分支，与代数变量（由多项式方程定义的几何对象）的几何性质有关。代数变化的一个简单例子是复杂的环面，它具有甜甜圈的形状。虽然在数学的其他分支（如拓扑学）中，所有的甜甜圈都具有相同的形状，但在代数几何中，精确的形状（在这种情况下是直径和宽度之间的比例）非常重要。事实上，在给定的拓扑类中精确量化几何物体的形状是模理论的主题。模理论是代数几何研究的一个中心领域，在数学和现代物理学中有许多应用。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The principal investigator is interested in the study of the geometry of moduli spaces, especially those that are birational to modular varieties of orthogonal or unitary type (examples include the moduli space of K3 surfaces or low genus curves). Laza's work exploits the existence of multiple birational models for a moduli space (e.g. obtained by using different compactification methods, such as Geometric Invariant Theory (GIT) or Hodge theory) to extract useful geometric information about a given moduli space. The principal investigator plans to apply this type of ideas to various projects involving moduli spaces. A first question that the PI proposes to investigate is the problem of finding a geometric compactification for the moduli of polarized K3 surfaces. The methods of the variation of GIT quotients and ideas coming from the minimal model program offer a promising approach to the low degree cases. A second project addresses various questions about the moduli of compact hyperkaehler manifolds, higher dimensional analogues of the K3 surfaces. In particular, the PI plans to investigate from an arithmetic and geometric point of view the case of moduli space of double EPW sextics (introduced by O'Grady). A third project is concerned with some concrete questions about the birational geometry of moduli spaces of genus 4 curves and cubic threefolds. The general area of the proposal is algebraic geometry, the branch of mathematics that is concerned with the geometric properties of algebraic varieties (geometric objects defined by polynomial equations). A simple example of algebraic variety is the complex torus, that has the shape of a doughnut. While in other branches of mathematics (such as topology) all doughnuts have the same shape, in algebraic geometry the precise shape (in this case the ratio between the diameter and width) is very important. In fact, the precise quantification of the shape of the geometric objects within a given topological class is the subject of the moduli theory. Moduli theory is a central field of study in algebraic geometry and has numerous applications in mathematics and modern physics.