|Speaker:||Jorgen Rennemo (ASC Oxford)|
|Title:||Homological projective duality for Sym^2 P^n|
|Date:||Tue, Jan 12, 2016, 13:15 - 14:45|
|Place:||Seminar Room A|
A famous theorem of Bridgeland says that if X and Y are Calabi-Yau 3-folds, then "X birational to Y" implies "the derived categories of X and Y are equivalent". The converse does not hold, and a few years ago Hosono and Takagi gave one example, when they proved that a certain pair of Calabi-Yau 3-folds X and Y have equivalent derived categories, but are not birational. Only a handful of such pairs are known.
In their example, X is a complete intersection of divisors in Sym^2(P^4). The derived category of such a complete intersections may often be understood via Kuznetsov's theory of homological projective duality. With Hosono & Takagi's example as motivation, we find a description of the "homological projective dual" of Sym^2 (P^n) for any n. As a corollary, this gives a partial computation of derived categories of complete intersections in Sym^2(P^n), and in particular recovers Hosono & Takagi's theorem when n = 4. I will explain this result and its proof, which is based on rephrasing the problem in terms of categories of matrix factorisations and then applying a variation of GIT stability.