Rencently a breakthrough was made in the context of the Minimal Model Program by the work of Birkar-Cascini-Hacon-McKernan. They proved that the canonical ring of a smooth or mildly singular projective algebraic variety is finitely generated.
Since I'm a master student and so I have no a wide view of the subject (I'm not an expert), I would like to know what are the main open problems in this direction (I mean, in the framework of the Mori Program). More generally, right now what are the driving forces, the big open questions in birational geometry?
Feel free to close this question, if too generic for the purposes of the site.
Thanks in advance.
Best Answer
[Just 'cause Artie asked:] :)
Many parts of the mmp are not know for log canonical pairs. There are many results in that direction, but also many questions are open. In some sense log canonical is a more natural class than klt or even dlt and it is very important from the point of view of applications to moduli theory because semi log canonical (the non-normal version of lc) singularities are closed under stable degeneration while klt singularities are not. A major difficulty stems from the fact that lc singularities are not rational.