In this paper we will describe an approach to mirror symmetry for appropriate one-dimensional DM stacks of arithmetic genus g ≤ 1, called tcnc curves, which was developed by the author with Treumann and Zaslow in Sibilla et al. (Ribbon Graphs and Mirror Symmetry I, arXiv:1103.2462). This involves introducing a conjectural sheaf-theoretic model for the Fukaya category of punctured Riemann surfaces. As an application, we will investigate derived equivalences of tcnc curves, and generalize classic results of Mukai on dual abelian varieties (Mukai, Nagoya Math. J. 81, 153–175, 1981).
