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Seminar

Fellowship of the Ring, National Seminar: Extremal Singularities in Prime Characteristic May 13, 2021 (01:30 PM PDT - 03:00 PM PDT)
Parent Program: --
Location: SLMath: Online/Virtual
Speaker(s) Karen Smith (University of Michigan)
Description

To attend this seminar, you must register in advance, by clicking HERE.

Keywords and Mathematics Subject Classification (MSC)
Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC
Video

Extremal Singularities in Prime Characteristic

Abstract/Media

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 "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.

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Extremal Singularities in Prime Characteristic