We state a number of conjectures that together allow one to classify a broad class of del Pezzo surfaces with cyclic quotient singularities using mirror symmetry. We prove our conjectures in the simplest cases. The conjectures relate mutation-equivalence classes of Fano polygons with Q-Gorenstein deformation classes of del Pezzo surfaces.where x, y, z are the standard co-ordinate functions on C 3 . Write w = mr + w 0 with 0 ≤ w 0 < r. It is known [24,25] that the base of the miniversal qG-deformation 2 of 1 n (1, q) is isomorphic to C m−1 and, choosing co-ordinate functions a 1 , . . . , a m−1 on it, the miniversal qG-family is given explicitly by the equation:xy + (z rm + a 1 z r(m−2) + · · · + a m−1 )z w0 = 0 ⊂ 1 r (1, w 0 a − 1, a) × C m−1 We say that 1 n (1, q) is of class T or is a T -singularity if w 0 = 0, and that it is a primitive T -singularity if w 0 = 0 and m = 1. T -singularities appear in the work of Wahl [28] and Kollár-Shepherd-Barron [25]. We say that 1 n (1, q) is of class R or is a residual singularity if m = 0, that is, if w = w 0 . We say that the singularity

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In earlier joint work with our collaborators Akhtar, Coates, Corti, Heuberger, Kasprzyk, Prince and Tveiten, we gave a conjectural classification of a broad class of orbifold del Pezzo surfaces, using Mirror Symmetry. We proposed that del Pezzo surfaces X with isolated cyclic quotient singularities such that X admits a Q-Gorenstein toric degeneration correspond under Mirror Symmetry to maximally mutable Laurent polynomials f in two variables, and that the quantum period of such a surface X, which is a generating function for Gromov-Witten invariants of X, coincides with the classical period of its mirror partner f .In this paper, we prove a large part of this conjecture for del Pezzo surfaces with 1 3 (1, 1) singularities, by computing many of the quantum periods involved. Our tools are the Quantum Lefschetz theorem and the Abelian/non-Abelian Correspondence; our main results are contingent on, and give strong evidence for, conjectural generalizations of these results to the orbifold setting.1991 Mathematics Subject Classification. X.

In this note we illustrate the Fanosearch programme of Coates, Corti, Galkin, Golyshev, and Kasprzyk in the example of the anticanonical cone over the smooth del Pezzo surface of degree 6. 1 arXiv:1901.06155v1 [math.AG]

We present a combinatorial criterion on reflexive polytopes of dimension 3 which gives a local-to-global obstruction for the smoothability of the corresponding Fano toric threefolds. As a result, we show an example of a singular Gorenstein Fano toric threefold which has compound Du Val, hence smoothable, singularities but is not smoothable.