The homological projective dual of Sym^2(P^n)

We produce semiorthogonal decompositions of the derived categories of complete intersections in Sym^2(P^n). These fit into Kuznetsov's theory of homological projective duality, which gives a description of the derived categories of all complete intersections in terms of the "homological projective dual" of Sym^2(P^n). An alternative approach to HP duality has recently been developed based on categories of matrix factorisations and VGIT. I will explain the necessary background for this method and show how it applies to our example. As an application of the HP duality statement, we reprove a theorem of Hosono and Takagi which says that a certain pair of non-birational Calabi-Yau 3-folds are derived equivalent.