Given two algebraic varieties $X,Y$ with finitely generated class group,
such that exist a small modification $\phi : X \rightarrow Y$,
i.e. there exists open subsets $U \subset X, V\subset Y$ such that 
$codim (X-U), codim (Y-V) \geq 2$,
$\phi (U) \subset V $ and $\phi |\_U : U \rightarrow V$ is an isomorphism.


Then there exists a equivalence relation on algebraic varieties given by 
$ X \sim Y \Leftrightarrow \text{ exist a small modification } \phi : X\dashrightarrow Y$.


Where i can find material about this relation?


Thanks in advance.



Small birational maps on algebraic varieties
If $X$ and $Y$ are smooth and proper, then this looks a lot like the $K$-equivalence, which states that for some (=any) smooth proper $Z$ that maps to both $X$ and $Y$ the exceptional divisors are the same, with multiplicities. You condition seems a bit weaker in that the exceptional divisors for the maps must only have the same support, but might have different coefficients.


$K$-equivalence is conjectured to imply equivalence of derived categories,
see <http://arxiv.org/abs/math/0205287>.


