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Seminar by Brent Doran (ETH Zürich)

Title: Homotopic replacement across geometry

2016.04.29 | Jane Jamshidi

Date Tue 24 May Wed 25 May
Time 16:15    17:15
Location 1531-215 (Aud. D3)


Toric varieties provide a model for many geometric ideas: from encoding orbit closures to wall-crossing in birational geometry, from theories of hypergeometric functions to effective counting of rational points on varieties over number fields. One way to generalize some important features of toric varieties is to study the Cox ring of a variety. Unfortunately, the Cox ring can be very difficult to work with, especially since it is rarely finitely generated (i.e., "Mori Dream spaces" are very rare). We present a general method for translating questions about generators of the Cox ring, and the corresponding divisors, into invariant theory and geometric invariant theory for the "homotopic replacement" space. This allows one to study previously inaccessible questions about effective cycles and rational point counts of bounded height, even when a space is far from being Mori Dream. We will present concrete applications in the context of some familiar spaces, including the Deligne-Mumford compactifications of moduli spaces of ordered and unordered points on spheres.