Abstract. Given a Laurent polynomial f , one can form the period of f : this is a function of one complex variable that plays an important role in mirror symmetry for Fano manifolds. Mutations are a particular class of birational transformations acting on Laurent polynomials in two variables; they preserve the period and are closely connected with cluster algebras. We propose a higher-dimensional analog of mutation acting on Laurent polynomials f in n variables. In particular we give a combinatorial description of mutation acting on the Newton polytope P of f , and use this to establish many basic facts about mutations. Mutations can be understood combinatorially in terms of Minkowski rearrangements of slices of P , or in terms of piecewise-linear transformations acting on the dual polytope P * (much like cluster transformations). Mutations map Fano polytopes to Fano polytopes, preserve the Ehrhart series of the dual polytope, and preserve the period of f . Finally we use our results to show that Minkowski polynomials, which are a family of Laurent polynomials that give mirror partners to many three-dimensional Fano manifolds, are connected by a sequence of mutations if and only if they have the same period.

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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We show that Fano lattice polygons define a class of balanced quivers with interesting properties. The combinatorics of these quivers is related to singularities of the underlying toric Fano surface. This allows us to show that every Fano polygon defines a point on a certain family of algebraic hypersurfaces. Our quivers admit a generalized mutation which preserves balancing and coincides with combinatorial mutation of Fano polygons whenever both operations are defined. We characterize balanced quivers arising from Fano polygons and discuss generalizations to higher dimensions.