Homological Mirror Symmetry for orbifold log Calabi-Yau surfaces

We construct mirror abstract Lefschetz fibrations associated to a class of surfaces with cyclic quotient singularities which we call effective. These surfaces can be obtained by contracting disjoint chains of smooth rational curves inside the anticanonical cycle $D$ of a smooth log Calabi-Yau surface $(Y,D)$ with maximal boundary and considering the result as an orbifold. The Fukaya-Seidel categories of these abstract Lefschetz fibrations admit semiorthogonal decompositions akin to the ones described via the derived special McKay correspondence of Ishii and Ueda arXiv:1104.2381v2 [math.AG]. We apply this construction to establish an equivalence at the large volume limit between the derived category of an effective orbifold log Calabi-Yau surface with points of type $\frac{1}{k}(1,1)$ and the Fukaya-Seidel category of its mirror Lefschetz fibration. We also compare the abstract construction to an explicit Landau-Ginzburg model defined by a Laurent polynomial associated to a toric degeneration in the case of the family of hypersurfaces $X_{k+1}\subset \mathbb{P}(1,1,1,k)$. The hypersurfaces $X_{k+1}$ admit a non-trivial moduli of complex structures, which we compare with an open subset of the space of symplectic structures on the total space of the mirror Landau-Ginzburg model via a mirror map built out of intrinsic quantities in a non-exact Fukaya-Seidel category.