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

Let $X=$\mathbb{P}$($\mathcal{O}${$\mathbb{P}$^s} \oplus \bigoplus{i=1}^r $\mathcal{O}$_{$\mathbb{P}$^s}(a_i))$beaFanoprojectivebundleover$$\mathbb{P}$^s$anddenoteby$Crit(X) \subset ($\mathbb{C}$^{\ast})^n$ the solution scheme of the Landau-Ginzburg system of equations of $X$. We describe a map $E:Crit(X)\to Pic(X)$ whose image $$\mathcal{E}$= {E(z) | z \in Crit(X) }$isthefullstronglyexceptionalcollectionon$X$ 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 $Crit(X)$.