Abstract:
We will explain the proof of the following theorem of Kontsevich and Tschinkel. Let $X\to B$ and $X'\to B$ be smooth and proper morphisms to a smooth connected curve $B$ over a field of characteristic zero. Suppose that their fibers over the generic point are birational (over the function field of $B$). Then their fibers over any closed point $b\in B$ are birational (over the residue field at $b$).
