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I had a minor thought. Did you ever look the paper Multiplicity of the special fiber of blowups by Corso,Polini, Vasconcelos.

In particular, they give an upper bound on the number (I realize that you are interested in lower bounds, but maybe some of the ideas are related / or could be useful) by in particular, bounding the multiplicity at the origin of the Rees algebra fiber ring (in other words, if you compute the blow-up by computing Proj R[It], $R[It]$, then mod out by an ideal from $R$, you get some graded ring corresponding to the fiber over the ideal you modded. Then you can study the blow-up by studying properties of the ring , and I think the multiplicity they study should give you an upper bound on the number of components).

Of course, the number of minimal associated primes gives you some bound on the components of the pre-image of $Z$.

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I had a minor thought. Did you ever look the paper Multiplicity of the special fiber of blowups by Corso,Polini, Vasconcelos.

In particular, they give an upper bound on the number (I realize that you are interested in lower bounds, but maybe some of the ideas are related / or could be useful) by in particular, bounding the multiplicity at the origin of the Rees algebra fiber ring (in other words, if you compute the blow-up by computing Proj R[It], then you can study the blow-up by studying properties of the ring, I think the multiplicity they study should give you an upper bound on the number of components).

Of course, the number of minimal associated primes gives you some bound on the components of the pre-image of $Z$.