  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 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.
