On Mirror Symmetry for (singular) del Pezzo surfaces out of the known constructions

Mirror symmetry predicts a correspondence between the complex geometry (the B side) and the symplectic geometry (the A side) of certain pairs of objects. In this talk I will discuss some aspects of mirror symmetry for Fano varieties, focusing on certain orbifold del Pezzo surfaces falling out of the standard mirror constructions. In particular, motivated by Homological Mirror Symmetry, I will describe the derived category of the surfaces (their B side), and mention some results on the category encoding the A side of their only-known Hodge-theoretic mirrors. This is based on work with A. Corti and work in progress with F. Rota.