Two-dimensional birational geometry and factorization centers

Using results of Manin-Iskovskikh on classification of geometrically rational surfaces over a perfect field, and results of Iskovskikh on classification of links between such surfaces, I will explain the proof for uniqueness of factorization centers in dimension two. Explicitly, the result is that the sequence of centers blown up and blown down, for any birational isomorphism $\phi: X\to Y$ is independent of $\phi$.