Minimal Model Program Learning Seminar:
Starting on October 27th 2026, Behrouz Taji, Chris Zeng, Junjie Chen and myself will be running a learning seminar on the minimal model program in algebraic geometry. A rough outline of topics to be discussed is as follows:
the cone and contraction theorems for surfaces;
the classification of complex projective surfaces with no (-1)-curves;
some general birational geometry (Iitaka map, Kodaira dimension, etc.);
the abundance theorem for surfaces;
singularities of the MMP with a focus on the case of surfaces;
the nonvanishing, rationality and basepoint freenes theorems.
We will (tentatively) proceed fortnightly from the 27th. Let me know if you want to be added to the mailing list.
Location: H13 4082
Schedule: 3-5PM Tuesday 27 Oct. Forward dates/times TBD.
October 27, 2026: The cone and contraction theorems for surfaces, Dominic Matan (UNSW)
Abstract: Surfaces are hard to classify up to isomorphism, not least because there are often no non-constant morphisms between them at all. We motivate birational classification instead, and then introduce the minimal model program for surfaces: first naively, by contracting $(-1)$-curves, and then through the cone of curves. Time permitting, we prove the cone and contraction theorems using the rationality and basepoint-freeness theorems. Finally, we explain how extremal contractions lead to a minimal model or a Mori fibre space as a “best” model in each birational class.
We will use, for the first lecture, my own notes together with Brendan Hassett's notes Rational surfaces over nonclosed fields and Kollár-Mori Birational geometry of algebraic varieties. I will upload my own notes following the lecture.