Hilbert-Kunz multiplicity/Monsky-Quartic for weird deformation behavior/example/short description/en

Let

$g=z^4+z^{2}xy+z(x^3+y^3)+(t+t^2)x^2{y}^2.$

Consider

$S={𝔽}_2[t,x,y,z]/(g).$

Then Monsky proved the following results on the Hilbert-Kunz multiplicity of the maximal ideal $(x,y,z)$ in $S{\otimes }_{{𝔽}_2[t]}L$, $L$ a field:

$e_{HK}(S{\otimes }_{{𝔽}_2[t]}L)=\{\begin{array}{c}3\text{ for }L={𝔽}_2(t)\\ 3+\frac{1}{4^d}\text{ for }L={𝔽}_q={𝔽}_2(\alpha ),(t\mapsto \alpha ,d=\mathrm{deg}(\alpha )).\end{array}$