Small birational maps on algebraic varieties

Question

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.

Answer 1

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.

Answer 2

I kind of doubt that you will find much written on this in this generality.

However, there are some special cases that might be of interest:

1. Lev already mentioned $K$-equivalence. As he remarks your case is more general. In any case the approach that could lead one to think about $K$-equivalence may be useful: By the Weak Factorization Theorem of Abramovich, Haru, Matsuki and Włodarczyk your birational map is a sequence of blow ups and blow downs with smooth centers. In order for that to be "small" the exceptional divisors of the blow ups and blow downs have to match up. You might get some mileage out of that.
2. A very important special case of small maps is flips. Flips are central to higher dimensional birational geometry, especially to the minimal model program. I would guess that this is your best bet. To start in this area look in Kollár-Mori's book. If that still does not give you enough, try to combine the techniques of the two references I gave.