Consider a $\textbf{flat}$ surjective map $f: X \rightarrow \mathbb{A}^1$. The general fibers $F_{\epsilon}$ are canonically isomorphic, and the special fiber $F_0$ above $0 \in \mathbb{A}^1$ is not isomorphic to the general fibers.

Given a closed subscheme $B \subset X$, we define its special fiber limit $\widetilde{B}$ to be the intersection of the special fiber $F_0$ with the closure of $B \times \mathbb{A}^1 \backslash \{ 0 \}$ in $X$, $F_0 \cap \overline{B \times \mathbb{A}^1 \backslash \{ 0 \}}$.

Let $B_1$ and $B_2$ be two closed subschemes of $X$. When does the limit of their intersection equal to the intersection of their limits, i.e. $\widetilde{B_1} \cap \widetilde{B_2} = \widetilde{B_1 \cap B_2}$? Is it enough for the dimension of $\widetilde{B_1} \cap \widetilde{B_2}$ to be equal to the dimension of $B_1 \cap B_2$? Any relevant comments and references are welcome. Thanks!

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    $\begingroup$ I don't understand the status of your second sentence. Is this a condition or a claim? If it's a claim, it sounds like you're missing some hypotheses. "Flat surjective map to $\mathbb{A}^1$" doesn't distinguish $0$ from any other point. $\endgroup$ – Qiaochu Yuan Mar 3 '16 at 18:33

No, your proposed condition is not really relevant.

Just consider the case where X is $\mathbb A^1 \times \mathbb A^1,$ with coords. $x$ and $y$, and projection to $x$ is the morphism to $\mathbb A^1$.

Now take $B_1$ to be the line $y = 0$, and take $B_2$ to be the line $xy.$ Then $\widetilde{B}_1 = \{(0,0)\},$ and $\widetilde{B}_2 = \{(0,0)\}$, and so $\widetilde{B}_1 \cap \widetilde{B}_2 = B_1\cap B_2 = \{(0,0)\}$. On the other hand, $\widetilde{B_1\cap B_2} = \emptyset$.

The key issue that underlies your question is, for $B_1$ and $B_2$ closed in $X$ and flat over $\mathbb A^1$, whether any point lying in $B_1\cap B_2 \cap F_0$ can be obtained as the specialization of a point in $(B_1\cap B_2 )\setminus F_0,$ or, more or less equivalently, whether $B_1\cap B_2$ is again flat over $\mathbb A^1$. And this obviously depends very much on the particular situation.


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