Singularities in Arbitrary Characteristic and Positivity

该项目涉及代数几何和交换代数中的问题。代数几何是研究代数簇的学科，代数簇是多项式方程组的解集。例如，在xy平面中，y=0的解由沿x轴的所有点组成，而xy=0的解由沿两个坐标轴的所有点组成。由于对于由xy=0定义的代数簇，在原点(0，0)处的切线没有定义，所以我们说这个簇在原点(0，0)处有奇点。本项目的第一个目标是进一步发展代数簇和更一般对象的奇点理论。第二个目标是建立这个研究项目所需的工具和技术，它将应用于代数几何和交换代数中的其他基本开放问题。研究将研究任意域上代数簇的奇异性，或者更广泛地说，任意Notherian环和方案的奇异性。即使主要感兴趣的是非奇异复代数簇，在这种更一般的背景下工作也往往是不可避免的。本项目的一个主要关注点是建立双曲面几何所需的基础和任意特征方案的最小模型程序。在前人的工作中，PI证明了Kodaira型消失定理适用于任意维等特征零点的格式，并发展了新的方法来取代正特征的Kodaira型消失定理。在本项目中，PI将应用这项先前工作中的技术来研究平坦态射下奇点的行为。PI还将研究不同类型奇点的附加类型结果的倒置。最后，PI计划扩展正特征技术来研究任意域上代数簇上的线丛的正性。这一奖项反映了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 project concerns questions in algebraic geometry and commutative algebra. Algebraic geometry is the study of algebraic varieties, which are solution sets for systems of polynomial equations. For example, in the xy-plane, the solutions for y=0 consist of all points along the x-axis, while the solutions for xy=0 consist of all points along both coordinate axes. Since the tangent line at the origin (0,0) is not defined for the algebraic variety defined by xy=0, we say that this variety has a singularity at the origin (0,0). The first goal of the present project is to further develop the theory of singularities of algebraic varieties and more general objects. The second goal is to build the tools and techniques necessary for this research program, which would have applications to other fundamental open questions in algebraic geometry and commutative algebra.The research will investigate singularities of algebraic varieties over arbitrary fields, or more generally of arbitrary Noetherian rings and schemes. Even when the primary interest is in non-singular complex algebraic varieties, working in this much more general context is often unavoidable. A major focus in the present project is to build the foundations necessary for birational geometry and the minimal model program for schemes of arbitrary characteristic. In previous work, the PI proved that Kodaira-type vanishing theorems hold for schemes in equal characteristic zero of arbitrary dimension and developed new methods to replace Kodaira-type vanishing theorems in positive characteristic. In the present project, the PI will apply the techniques from this previous work to study the behavior of singularities under flat morphisms. The PI will also investigate inversion of adjunction-type results for various classes of singularities. Finally, the PI plans to extend techniques in positive characteristic to study positivity of line bundles on algebraic varieties over arbitrary fields.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.