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

Abstract: Let $X=\mathbb{P}(\mathcal{O}{\mathbb{P}^s} \oplus \bigoplus{i=1}^r \mathcal{O}_{\mathbb{P}^s}(a_i))$ 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 $E : Crit(X) \rightarrow Pic(X)$ whose image $\mathcal{E}= {E(z) | z \in Crit(X) }$ is the full strongly exceptional collection on $X$ found by Costa and Mir$\acute{\textrm{o}}$-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 $Crit(X)$.