Extremal Surfaces in Positive Characteristic

What is the most singular possible point on any variety in positive characteristic?  In joint work with coauthors, we gave a precise answer to this question for hypersurfaces using a measure of singularity called the F-pure threshold, and we called these “most singular” hypersurfaces extremal hypersurfaces.  These special hypersurfaces only occur in degrees $d=p^e+1$, where $p$ is the characteristic.  In each degree where they occur, there is one particular extremal hypersurface (up to a change of variables) which stands out—it is the cone over a smooth projective hypersurface.  In this talk, we’ll focus on this special hypersurface in the 4-variable case—that is, we’ll focus on an extremal hypersurface which is the cone over a smooth projective surface, and we’ll discuss some of its surprising geometric properties.  In particular, we highlight a conjecture which further justifies the name “extremal,” which would answer a longstanding classical question about smooth projective surfaces of any degree.

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