X^p+Y^(p+1)+Z^(p+1)/Signaturen/Beispiel/en

We consider in characteristic ${}p$ the ring given by the equation

${}X^{p}+Y^{p+1}+Z^{p+1}=0\,.$

This is a normal hypersurface ring. This ring is not ${}F$-pure, since ${}X\notin (Y,Z)$, but ${}X^{p}\in (Y^{p},Z^{p})$. Hence it is not strongly ${}F$-regular and so its ${}F$-signature is ${}0$. However, the differential signature is positive. This example is also an easy counterexample to the Zariski-Lipman conjecture.