In this thesis we discuss some aspects of Mirror Symmetry for Fano varieties and toric singularities. We formulate a conjecture that relates the quantum cohomology of orbifold del Pezzo surfaces to a power series that comes from Fano polygons. We verify this conjecture in some cases, in joint work with A. Oneto.
We generalise the Altmann–Mavlyutov construction of deformations of toric singularities: from Minkowski sums of polyhedra we construct deformations of affine toric pairs. Moreover, we propose an approach to the study of deformations of Gorenstein toric singularities of dimension 3 in the context of the Gross–Siebert program.
We construct deformations of polarised projective toric varieties by deforming their affine cones. This method is explicit in terms of Cox coordinates and it allows us to give explicit equations for a construction, due to Ilten, which produces a deformation between two toric Fano varieties when their corresponding polytopes are mutation equivalent. We also provide examples of Gorenstein toric Fano 3-folds which are locally smoothable, but not globally smoothable.