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Is the local ring $R_p$ reduced, where $p=(a,x,y,z)$ and $R=k[a,x,y,z]/I$ and $I=(a^2(x+1)-z^2,ax(x+1)-yz,xz-ay,y^2-x^2(x+1))$ ?

This comes from example III.9.8.4 in Hartshorne's algebraic geometry. There he explains that $R/(a)$ has a non-reduced local ring at $p=(x,y,z)$.

My question is motivated by the following (all reference numbers are from Hartshorne):

  • I think that $X=Spec R$ is not reduced at $(0,0,0,0)$ (corresponding to $p$).
  • There is a map $X_{red}\to X$ that gets rid of the non-reduced structure.
  • The composition $X_{red}\to X\to Spec\quad k[a]$ should be the family he mentions in example III.9.10.1.
  • III.9.7 says that a reduced and irreducible family X over a smooth Y is flat (X dominating Y).
  • The last two bullets contradict each other since example III.9.10.1 precisely says that this family is not flat.

Any help in sorting out where the problem in my reasoning is welcome!

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