Commutative Algebra: Extremal Singularities in Prime Characteristic

该项目是代数几何学，这是一门理解可以用多项式方程描述的几何形状的科学。多项式在数学中无处不在，因为它们可以很容易地用手或机器操作，但它们表现出的行为范围很广，几乎可以模拟任何形状，从简单的圆和线到可以出现在医学成像或动画电影中的复杂图像。该项目试图理解代数变体的奇点-形状被捏或折叠的地方。我们将设计工具来衡量这些奇点有多“糟糕”，并尝试对“最糟糕”的奇点进行分类。这个项目将由PI与一组实习生一起进行，包括本科生、研究生、博士后和其他合作者。该项目研究特征为p的代数闭域上多项式的F-纯门限的下界。目标是找到精确的下界，对达到该下界的奇点进行分类，并将这些结果应用于该领域的公开问题。更具体地说，第一个方向是根据其他不变量，如重数，证明F-纯阈值的一般下界，然后识别达到这些下界的变种--“极端奇点”。接下来，我们的目的是对Frobenius形式进行分类，它是齐次情形下F-纯阈值的某个下界的“极值奇点”。第三个方向是推进Kleiman和Piene猜想的进展，利用Frobenius形式的性质刻画其Gauss映射在某些方面是极值的超曲面。最终的方向是调查Frobenius形式是否可以定义允许非对易奇点解的超曲面，这与之前关于具有非对易解的变种的奇点温和的猜测相反。这一裁决反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

The project is in algebraic geometry, the science of understanding geometric shapes that can be described by polynomial equations. Polynomials are ubiquitous in mathematics because they can be easily manipulated by hand or by machine, yet they exhibit a large range of behavior which can model nearly any shape, from simple circles and lines to complicated images which could appear in medical imaging or an animated movie. The project seeks to understand the singularities of algebraic varieties---places where the shape is pinched or folded over on itself. We will devise tools to measure how "bad" these singularities are, and attempt to classify the "worst" ones. The project will be carried out by the PI with a team of trainees, including undergraduate students, graduate students, post-docs, and other collaborators.The project investigates lower bounds on the F-pure threshold of polynomials over an algebraically closed field of characteristic p. The goal is to find sharp lower bounds, classify the singularities achieving that lower bound, and apply these results to open problems in the field. More specifically, a first direction is to prove general lower bounds on F-pure threshold in terms of other invariants, such as multiplicity, and then identify the varieties—“extremal singularities”— achieving those bounds. Next, the aim is to classify Frobenius forms, which are the “extremal singularities” for a certain lower bound on F-pure threshold in the homogeneous case. A third direction is to advance progress on a conjecture of Kleiman and Piene characterizing hypersurfaces whose Gauss maps are extremal in certain ways, using properties of Frobenius forms. The final direction is an investigation into whether Frobenius forms may define hypersurfaces admitting non-commutative resolutions of singularities, contrary to previous speculations about mildness of singularities for varieties with non-commutative resolutions.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.