Extremal Singularities in Positive Characteristic

We prove a general lower bound on the $F$-pure threshold of a reduced form of characteristic $p>0$ in terms of its degree, and investigate the class of forms that achieve this minimal possible $F$-pure threshold. Specifically, we prove that if $f$ is a reduced homogenous polynomial of degree $d$, then its $F$-pure threshold (at the unique homogeneous maximal ideal) is at least $\frac{1}{d-1}$. We show, furthermore, that its $F$-pure threshold equals $\frac{1}{d-1}$ if and only if $f\in \mathfrak m^{[q]}$ and $d=q+1$, where $q$ is a power of $p$. Up to linear changes of coordinates (over a fixed algebraically closed field), we show that there are only finitely many such "extremal singularities" of bounded degree and embedding dimension, and only 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.