Fermat quartic/Hilbert-Kunz/Dependence on characteristic/Example

The Fermat quartic ${}x^{4}+y^{4}+z^{4}=0$ is the easiest example where the Hilbert-Kunz multiplicity of the maximal ideal ${}(x,y,z)$ fluctuates with the characteristic. We have

${}e_{HK}(R_{p})={\begin{cases}3{\text{ for }}p=\pm 1\,\operatorname {mod} \,8\\3+{\frac {1}{p^{2}}}{\text{ for }}p=\pm 3\,\operatorname {mod} \,8\,.\end{cases}}\,$

The limit is of course ${}3$, which corresponds to the fact that the syzygy bundle is semistable in characteristic zero. The syzygy bundle is semistable for all prime characteristics ${}p\geq 3$, but not strongly semistable for the prime numbers ${}p=\pm 3\,\operatorname {mod} \,8$.