Karen Smith (Oct. 9, 2020):  Extremal Singularities in Prime Characteristic
Channel: Algebraic Geometry at Stanford
Date: 2021-04-20

What is the most singular possible singularity? What can we say about its geometric and algebraic properties? This seemingly naive question has a sensible answer in characteristic $p$.
The "F-pure threshold," which is an analog of the log canonical threshold,  can be used to "measure" how bad a singularity is. The F-pure threshold is a numerical invariant of a point  on (say)  a hypersurface---a positive rational number that is 1 at any smooth point (or more generally, any F-pure point) but less than one in general, with "more singular" points having smaller F-pure thresholds. We explain a recently proved  lower bound on the F-pure threshold in terms of the multiplicity of the singularity. We also show that there is a nice class of hypersurfaces---which we call "Extremal hypersurfaces"---for which this bound is achieved. These have very nice (extreme!) geometric properties. For example, the affine cone over a non Frobenius split cubic surface of characteristic two is one example of an "extremal singularity". Geometrically, these are the only cubic surfaces with the property that *every* triple of coplanar lines on the surface meets in a single point (rather than a "triangle" as expected)---a very extreme property indeed.