When is $\operatorname{Proj}k[x,y,z]/(x^n＋y^n＋z^n)$, with $n\geq 1$ and $k$ an algebraically closed field, a regular scheme? From Liu p135, the answer is '$n$ is prime to $ch(k)$'.
I tired to use Jacobian criterion but I donnnot how to apply it. I would be appreciated if you could teach me how to apply Jacobi critearion, thank you.
Question: "When is $Proj(k[x,y,z]/(x^n＋y^n＋z^n))$, with $n≥1$ and $k$ an algebraically closed field, a regular scheme?"
Answer: If $char(k)=p>0$ and $n=lp$ it follows
$$F(x,y,z)=x^n+y^n+z^n=(x^l)^p+(y^l)^p+(z^l)^p=(x^l+y^l+z^l)^p$$
and then the polynomial $F(x,y,z)$ defines a non-reduced projective scheme, which is not regular.
If $(char(k),n)=1$, it follows a local check shows that $Z(F)$ is regular: As an example: At $D(x)$ you get the polynomial $f(u,v):=1+u^n+v^n$ with $u:=y/x, v:=z/x$, and $f_u:=nu^{n-1}, f_v:=nv^{n-1}$. It follows the ideal $(f,f_u,f_v)=(1)$ is the unit ideal, hence $Z(f)$ is regular. Similar for $D(y),D(z)$.