# Extremal Singularities in Prime Characteristic

*Karen Smith (University of Michigan)*

**09-Oct-2020, 19:00-20:00 (2 years ago)**

**Abstract: **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.

algebraic geometry

Audience: researchers in the topic

( video )

**Comments: **The discussion for Karen Smithâ€™s talk is taking place not in zoom-chat, but at tinyurl.com/2020-10-09-ks (and will be deleted after ~3-7 days).

**Stanford algebraic geometry seminar **

**Series comments: **This seminar requires both advance registration, and a password.
Register at stanford.zoom.us/meeting/register/tJEvcOuprz8vHtbL2_TTgZzr-_UhGvnr1EGv
Password: 362880

If you have registered once, you are always registered for the seminar, and can join any future talk using the link you receive by email. If you lose the link, feel free to reregister. This might work too: stanford.zoom.us/j/95272114542

More seminar information (including slides and videos, when available): agstanford.com

Organizer: | Ravi Vakil* |

*contact for this listing |