Commutative Algebra: Extremal Singularities in Prime Characteristic

该项目涉及代数几何，即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大，几乎可以模拟任何形状，从简单的圆和线到复杂的图像，可以出现在医学成像或动画电影中。该项目试图理解代数变体的奇异性——形状被挤压或折叠的地方。我们将设计工具来衡量这些奇点有多“坏”，并尝试对“最坏”的奇点进行分类。该项目将由PI与包括本科生、研究生、博士后和其他合作者在内的实习生团队进行。该项目研究了特征为p的代数闭域上多项式的f纯阈值的下界。目标是找到尖锐的下界，对达到该下界的奇点进行分类，并将这些结果应用于该域中的开放问题。更具体地说，第一个方向是证明f -纯阈值的一般下界，根据其他不变量，如多重性，然后确定变种-“极端奇点”-实现这些边界。其次，目的是分类Frobenius形式，它是在齐次情况下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的猜想，利用Frobenius形式的性质来描述高斯映射在某些方面是极值的超曲面。最后一个方向是研究Frobenius形式是否可以定义承认奇异点的非交换分辨率的超曲面，这与先前关于具有非交换分辨率的变种的奇异点的温和性的推测相反。该奖项反映了美国国家科学基金会的法定使命，并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，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.