Suppose $B=A\oplus S^1\oplus S^2\dots$ is a graded ring, $B$ is generated by $S^1$, $C=\textrm{Spec}B$ is called a cone over $X=\textrm{Spec}A$. We have natural projection $\pi\colon C\to X$. Moreover, we assume $A,B$ are coordinate rings of varieties. 


Suppose $F\_k$ is the locus of points in $X$ whose fiber dimension is exactly $k$. Suppose $Z$ is an irreducible component of $\overline{F\_k}$.


Is $\pi^{-1}(Z)$ an irreducible component of $C$?



Irreducible components of a cone
That is not true, and the blowing up of the origin in the plane is a counterexample: $A$ is $k[s,t]$, and $B$ is $k[s,t,u,v]/\langle ut-sv\rangle$. 


