On Landau-Ginzburg systems and $\mathcal{D}^b(X)$ of various toric Fano manifolds with small picard group

Abstract: For a toric Fano manifold $X$ denote by $Crit(X) \subset (\mathbb{C}^{\ast})^n$ the solution scheme of the Landau-Ginzburg system of equations of $X$. Examples of toric Fano manifolds with $rk(Pic(X)) \leq 3$ which admit full strongly exceptional collections of line bundles were recently found by various authors. For these examples we construct a map $E : Crit(X) \rightarrow Pic(X)$ whose image $\mathcal{E}=\left { E(z) \vert z \in Crit(X) \right }$ is a full strongly exceptional collection satisfying the M-aligned property. That is, under this map, the groups $Hom(E(z),E(w))$ for $z,w \in Crit(X)$ are naturally related to the structure of the monodromy group acting on $Crit(X)$.