= Nonreduced reducible fibre between integral schemes
A <morphism of schemes> between <integral schemes> can have a <scheme-theoretic fibre> that is neither reduced nor irreducible. For the map of <affine lines> given by $t=x^2(x-1)^2$, the fibre at $t=0$ has ring $k[x]/(x^2(x-1)^2)$. The <Chinese remainder theorem> identifies it with the product of two <dual number> rings, so it consists of two <nonreduced double points>. Its nilpotent structure is invisible to the set-theoretic fibre.
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