On Landau-Ginzburg Systems and ${\mathcal{D}}^b(X)$ of projective bundles

Abstract

Let $X=\mathbb{P}({\mathcal{O}}_{{\mathbb{P}}^s}\oplus {⨁}_{i=1}^r{\mathcal{O}}_{{\mathbb{P}}^s}(a_i))$ be a Fano projective bundle over ${\mathbb{P}}^s$ and denote by $Crit(X)\subset ({\mathbb{C}}^{\ast }{)}^n$ the solution scheme of the Landau-Ginzburg system of equations of $X$. We describe a map $E:Crit(X)\to Pic(X)$ whose image $\mathcal{E}=\{E(z)|z\in Crit(X)\}$ is the full strongly exceptional collection on $X$ found by Costa and Mir$\acute{\text{o}}$-Roig. We further show that $Hom(E(z),E(w))$ for $z,w\in Crit(X)$ can be described in terms of a monodromy group acting on $Crit(X)$.

17 pages, 3 figures

References in corpus (1)

• A note on the Frobenius morphism on toric varieties