Landau-Ginzburg models for Fano threefolds of Picard rank one and exceptional collections

Abstract: We study fibers with isolated singularities of Landau-Ginzburg models for Fano threefolds of Picard rank one. We compare the data we get with maximal known lengths of exceptional collections in derived categories of coherent sheaves on the Fano threefolds, verify some predictions of Homological Mirror Symmetry, and present some expectations about exceptional collections for Fano threefolds.