Seminar Algebra and Geometry: Jesus Martinez (MPIM Bonn)

Moduli space of cubic surfaces and their anti canonical divisors

We study variations of GIT quotients of log pairs (X,D) where X is a hypersurface of some fixed degree and D is a hyperplane section. GIT is known to provide a finite number of possible compactifications of such pairs, depending on one parameter. Any two such compactifications are related by birational transformations. We describe an algorithm to study the stability of the Hilbert scheme of these pairs, and apply our algorithm to the case of cubic surfaces. Finally, we relate these compactifications to the (conjectural) moduli space of logK-semistable pairs showing that any log K-stable pair is an element of our moduli and that there is a canonically defined CM line bundle that polarizes our moduli. This is a joint work with Patricio Gallardo (University of Georgia) and Cristiano Spotti (Aarhus University).

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