Session S07 - Differential operators in algebraic geometry and commutative algebra

Monday, July 19, 18:30 ~ 19:00 UTC-3

Lines on Extremal Surfaces

Karen Smith

University of Michigan, United States   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

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. In recent work with Anna Brosowski, Janet Page and Tim Ryan, we have unravelled the story of the lines on extremal surfaces of any degree.

Joint work with Anna Brosowksi (University of Michigan), Janet Page (University of Michigan) and Tim Ryan (University of Michigan).

View abstract PDF