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Let $f:X\dashrightarrow Y$ be the flip of a small contraction $\phi:X\rightarrow Z$, and let $\psi:Y\rightarrow Z$ be the small contraction such that $\psi\circ f = \phi$. Let $Exc(\phi), Exc(\psi)$ be the exceptional loci of $\phi$ and $\psi$ respectively.

Do we always have $\dim(Exc(\psi)) = \operatorname{codim}_X(Exc(\phi))-1$ ?

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I don't think so. In general we always have $$ \dim \text{Exc}\phi+\dim \text{Exc}\psi\geq \dim X-1. $$ This is proved in Lemma 5-1-7 [Kawamata, Matsuda, Matsuki, Introduction to Minimal Model Program].

But in general the equality may not hold. I did not come up with any example according to my knowledge, but I remember that at least in [KMM] they discussed about 4-dimensional filps of type $(2,2)$ (type means the dimension of exceptional loci of both sides).

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