LOWER BOUNDS ON THE F-PURE THRESHOLD AND EXTREMAL SINGULARITIES

We prove that if f is a reduced homogeneous polynomial of degree d, then its F-pure threshold at the unique homogeneous maximal ideal is at 1 least d−1^{. We show, furthermore, that itsF-pure threshold equals} 1 if and d−1 only if f ∈ m^[q] and d = q +1, where q is a power of p. Up to linear changes of coordinates (over a fixed algebraically closed field), we classify such “extremal singularities”, and show that there is at most one with isolated singularity. Finally, we indicate several ways in which the projective hypersurfaces defined by such forms are “extremal”, for example, in terms of the configurations of lines they can contain.