Birational geometry and effective theory of minimal models

The minimal model program in birational geometry solves the problem on the construction of adistinguished element in a given birational equivalence class of algebraic varieties. The two-dimensional theory was constructed in classical works of Castelnuovo and Enriques, and foundationsof its higher-dimensional analoguewere originated by M. Reid and S. Mori. Although at present the existence of minimal models fully established, the question of the effective implementation of the program is still open, even in dimension 3. In this talk, we discuss recent advances in this direction.
The report is based on joint works with Shigefumi Mori.