Let $p:X\to\mathbb A^3$ be the blow-up of $\mathbb A^3$ along a line $L$. Denote the exceptional divisor as $E$. Let $H$ be a plane containing $L$ and $\tilde{H}$ be the strict transform. I found contradiction when computing some intersection numbers.
Let $C=\tilde{H}\cap E$. Then projection formula implies that $$p^*H\cdot C=p_*(p^*H\cdot C)=H\cdot p_*C=H\cdot L=1.$$ The intersections are interpreted as intersections in the Chow ring of $X$ and $\mathbb A^3$.
On the other hand, however, $E\cdot C=0$ since $C$ deforms in $\tilde{H}$ in a ruling and a small perturbation of $C$ is disjoint from the exceptional divisor $E$. Similarly, $\tilde{H}\cdot C=0$ since $C$ can be deformed in $E$ (consider a family of planes $H_t$ containing $L$, their strict transforms on $E$ are disjoint) and a small deformation will be disjoint from $\tilde{H}$.
So $0=E\cdot C+\tilde{H}\cdot C=p^*H\cdot C=1$. Contradiction. Where is the mistake?