Homological Projective Duality

Abstract

This thesis is concerned with the theory of homological projective duality of A. Kuznetsov. The varieties of interest are projective bundles. By considering the resolution $X =\mathrm{Hilb}^2\mathbb{P}^2$ of the variety $\mathrm{Sym}^2\mathbb{P}^2$ as a projective bundle, we show using results of Kuznetsov that the homological projective dual $Y$ of $X$ agrees with that of the smooth stack $[\mathbb{P}^2\times \mathbb{P}^2/S_2]$ as described by J. Rennemo. Further, we describe the homological projective dual of a family of projective bundles over the Grassmanian $G(n,n+1)$ to which $X$ belongs. Lastly we study the duality of $X$ and $Y$ on linear sections and show an equivalence of derived categories of a pair of elliptic curves $X_L \subset X$ and $Y_L \subset Y$.