  Let $k$ be a field of characteristic $p > 0$ such that $[k:k^p] < \infty$ and
let $f \in R = k[x_0, ..., x_n]$ be homogeneous of degree $d$. We obtain a
sharp bound on the degrees in which the Frobenius action on
$H^n_\mathfrak{m}(R/fR)$ can be injective when $R/fR$ has an isolated
non-F-pure point at $\mathfrak{m}$. As a corollary, we show that if
$(R/fR)_\mathfrak{m}$ is not F-pure then $R/fR$ has an isolated non-F-pure
point at $\mathfrak{m}$ if and only if the Frobenius action is injective in
degrees $\le -n(d-1)$.
