  Let $(R, m)$ be a local ring of prime characteristic $p$ and $q$ a varying
power of $p$. We study the asymptotic behavior of the socle of $R/I^{[q]}$
where $I$ is an $m$ -primary ideal of $R$. In the graded case, we define the
notion of diagonal $F$-threshold of $R$ as the limit of the top socle degree of
$R/m^{[q]}$ over $q$ when $q \to \infty$. Diagonal $F$-threshold exists as a
positive number (rational number in the latter case) when: (1) $R$ is either a
complete intersection or $R$ is $F$-pure on the punctured spectrum; (2) $R$ is
a two dimensional normal domain. In the latter case, we also discuss its
geometric interpretation and apply it to determine the strong semistability of
the syzygy bundle of $(x^d, y^d,z^d)$ over the smooth projective curve in
$\mathbb P^2$ defined by $x^n+y^n+z^n=0$. The rest of this paper concerns a
different question about how the length of the socle of $R/I^{[q]}$ vary as $q$
varies. We give explicit calculations of the length of the socle of $R/m^{[q]}$
for a class of hypersurface rings which attain the minimal Hilbert-Kunz
function. We finally show, under mild conditions, the growth of such length
function and the growth of the second Betti numbers of $R/m^{[q]}$ differ by at
most a constant, as $q \to \infty$.

Commutative Algebra