May 13, 2021
Thursday

01:30 PM  03:00 PM


Fellowship of the Ring, National Seminar: Extremal Singularities in Prime Characteristic
Karen Smith (University of Michigan)

 Location
 MSRI: Online/Virtual
 Video

 Abstract
To attend this seminar, you must register in advance, by clicking HERE.
Abstract: What is the most singular possible singularity? What can we say about it's geometric and algebraic properties? This seemingly naive question has a sensible answer in characteristic p. The "Fpure threshold," which is an analog of the log canonical threshold, can be used to "measure" how bad a singularity is. The Fpure threshold is a numerical invariant of a point on (say) a hypersurfacea positive rational number that is 1 at any smooth point (or more generally, any Fpure point) but less than one in general, with "more singular" points having smaller Fpure thresholds. We explain a recently proved lower bound on the Fpure threshold in terms of the multiplicity of the singularity. We also show that there is a nice class of hypersurfaceswhich 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.
 Supplements

Notes
3.57 MB application/pdf


