Commutative Algebra: Extremal Singularities in Prime Characteristic

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

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.