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Algebraic Geometry Student's Seminar : What is...a flip?

Sándor Kovács, University of Washington
Monday, April 29, 2024 - 4:00pm to 5:00pm
CMU 228
An important part of higher dimensional geometry is the minimal model program. For surfaces, this is classical, going back to the famous 19th century Italian school of algebraic geometry. Early difficulties at generalizing the theory to higher (i.e., >2) dimensions first led people to believe that a higher dimensional theory is not possible. It was realized by Reid and Mori in the late 70s that a meaningful theory is in fact possible if one allows mild singularities on their birational models. The 3-dimensional theory was developed in the 80s and the general theory about 20 years later.
The starting point is a higher dimensional analogue of Castelnuovo's theorem on contractibility of (-1)-curves. In higher dimensions this leads to multiple possibilities and one of the possibilities seem to lead into a deadend, namely producing an image that has worse singularities than the original theorem could possibly handle. In order to get around this obstacle, another operation, named a *flip* was invented which allows the process to continue. 
I will try to explain how one gets to this point, what a flip is, and how it solves the problem at hand. 
Note that the existence of a flip is highly non-trivial. The proof of its existence earned Mori a Fields Medal in 1990 for the 3-dimensional case and Hacon and McKernan a Breakthrough Prize in 2018 for the general case.