Maximally mutable Laurent polynomials

We introduce a class of Laurent polynomials, called maximally mutable Laurent polynomials (MMLPs), which we believe correspond under mirror symmetry to Fano varieties. A subclass of these, called rigid, are expected to correspond to Fano varieties with terminal locally toric singularities. We prove that there are exactly 10 mutation classes of rigid MMLPs in two variables; under mirror symmetry these correspond one-to-one with the 10 deformation classes of smooth del Pezzo surfaces. Furthermore, we give a computer-assisted classification of rigid MMLPs in three variables with reflexive Newton polytope; under mirror symmetry these correspond one-to-one with the 98 deformation classes of three-dimensional Fano manifolds with very ample anti-canonical bundle. We compare our proposal to previous approaches to constructing mirrors to Fano varieties, and explain why mirror symmetry in higher dimensions necessarily involves varieties with terminal singularities. Every known mirror to a Fano manifold, of any dimension, is a rigid MMLP.

TC, 0000-0003-0779-9735; AMK, 0000-0003-2340-5257 We introduce a class of Laurent polynomials, called maximally mutable Laurent polynomials (MMLPs), which we believe correspond under mirror symmetry to Fano varieties. A subclass of these, called rigid, are expected to correspond to Fano varieties with terminal locally toric singularities. We prove that there are exactly 10 mutation classes of rigid MMLPs in two variables; under mirror symmetry these correspond one-to-one with the 10 deformation classes of smooth del Pezzo surfaces. Furthermore, we give a computer-assisted classification of rigid MMLPs in three variables with reflexive Newton polytope; under mirror symmetry these correspond one-to-one with the 98 deformation classes of threedimensional Fano manifolds with very ample anticanonical bundle. We compare our proposal to previous approaches to constructing mirrors to Fano varieties, and explain why mirror symmetry in higher dimensions necessarily involves varieties with terminal singularities. Every known mirror to a Fano manifold, of any dimension, is a rigid MMLP. between Fano manifolds and Laurent polynomials. An n-dimensional Fano manifold X determines the regularized quantum period G X (t) = 1 + ∞ k=2 k!c k t k .
(1.1) This is a generating function for certain Gromov-Witten invariants of X: roughly speaking, the coefficient c k is the number of degree-k rational curves on X that pass through a given point (see [1] for details). The quantum period is expected to characterize X. Mirror symmetry suggests that the power series G X also arises from a Laurent polynomial f ∈ C[x ±1 1 , . . . , x ±1 n ], and when this happens there is a close connection between the geometry of X and of f . More precisely, given a Laurent polynomial f ∈ C[x ±1 1 , . . . , x ±1 n ], one can form its classical period: The Taylor expansion of π f , which we also denote π f , can be readily calculated [2,3]: Here coeff 1 ( f k ) denotes the coefficient of the constant term of f k . If then we say that f is a mirror partner for X. This is a very strong condition on f . However, certain Laurent polynomials are known to arise as mirror partners to n-dimensional Fano manifolds [1,4,5], and in particular mirror partners exist for all Fano manifolds of dimension up to three. Henceforth let us consider only Laurent polynomials f such that the exponents of monomials in f generate the lattice Z n ; this will avoid a subtlety about finite coverings of toric varieties. If a Laurent polynomial f is a mirror partner to a Fano manifold X, then it is expected that X admits a Q-Gorenstein (qG-) degeneration to the singular toric variety X P associated with the Newton polytope P = Newt(f ) of f [6,7]. Here X P is given by taking the spanning fan, also called the face fan or central fan, whose cones span the faces of P. Given a mirror partner f , new mirror partners can be generated via mutation; this process, which produces new Laurent polynomials with the same classical period, is described in detail below. Suppose that f is a mirror partner to X. It is conjectured that the Laurent polynomial g is also a mirror partner to X if and only if f and g are connected via a sequence of mutations. Such a connection implies in particular that the corresponding toric varieties X P and X Q , P = Newt(f ), Q = Newt(g), are related via qG-deformation [8].

(a) Fano polytopes
We now attempt to reverse the construction described above. Let N be a lattice of rank n, and let P ⊂ N Q := N ⊗ Z Q be a convex lattice polytope. with coefficients c v such that Newt(f ) = P, with f a mirror partner to some Fano manifold?
Without loss of generality, we may make the following three assumptions.
(i) We may assume that the origin 0 of N is contained in the relative interior P • := P \ ∂P of P: if 0 / ∈ P then, for any Laurent polynomial f with Newt(f ) = P, the period π f will be constant, and hence cannot be the quantum period of a Fano manifold; if 0 ∈ ∂P then we can reduce to a lower-dimensional situation by considering the smallest dimensional face F of P with 0 ∈ F. (ii) We can assume that dim(P) = n, since otherwise we can restrict to the sublattice given by span Q (P) ∩ N.
of N generated by the lattice points of P. Restricting to the sublattice N will not change the corresponding period π f , since coeff 1 (f k ) depends on N only via the linear relations between the lattice points of P. Hence we may assume that N = N.
Recall that we want to associate with P a toric variety X P via the spanning fan. The rays of this fan are spanned by the vertices vert(P) of P, so we require that the vertices are primitive lattice points of N (and hence correspond to the lattice generators of the rays). By assumption (i), the fan is complete, and we see that X P is an n-dimensional toric Fano variety.

Definition 1.2.
A convex lattice polytope P ⊂ N Q is called Fano if it is of maximum dimension with respect to the ambient lattice N, if it contains the origin 0 in its interior P • , and if the vertices of P are primitive lattice elements.
A Fano polytope P corresponds to toric Fano variety X P , and-considering polytopes up to the action of GL(N), and toric varieties up to isomorphism-this correspondence is bijective. Our focus in this paper will be on Fano polytopes, however many of the combinatorial arguments generalize if we drop the requirement that the vertices are primitive; such polytopes correspond to toric Deligne-Mumford stacks (e.g. [9]). For an overview of Fano polytopes, see [10]. Assumption (iii), although essential when considering qG-deformations of the toric variety X P , is not part of the definition of Fano polytope. In two dimensions, every Fano polygon satisfies assumption (iii), since every Fano polygon contains a basis for the lattice; this is not true in higher dimensions. We have that |P ∩ N| = |Q ∩ N| = 6, with the only non-vertex lattice point in each case corresponding to the origin. The points of Q generate the sublattice e 1 · Z + e 2 · Z + 3e 3 · Z of index three, and restricting Q to this sublattice gives P. The Laurent polynomials which have Newton polytopes P and Q, respectively, generate the same period sequence: This agrees with the period sequence for P 1 × P 2 given by Givental [11], and in fact the toric variety X P is equal to P 1 × P 2 . The anti-canonical degree of X P is (−K X P ) 3 = Vol(P * ) = 54, where is the dual (or polar) polytope to P in the dual lattice M := Hom(N, Z), and Vol(·) denotes the latticenormalized volume. The degree of X Q differs from that of X P by a factor of three, that being the index of the sublattice, so that (−K X Q ) 3 = Vol(Q * ) = 1/3 Vol(P * ) = 18. In particular, even though π f = π g = G P 1 ×P 2 , X Q cannot be a qG-degeneration of P 1 × P 2 .  [12,13]. Since P is a smooth Fano polytope, the corresponding toric variety X P is a Fano manifold. Any toric Fano variety can be expressed as a GIT quotient of the form C r / /(C × ) k -this generalizes the construction of projective space P r−1 as (C r \{0})/C × = C r / /C × , and amounts to specifying 1 the characters B 1 , . . . , B r of (C × ) k that define the action of (C × ) k on C r . The quantum period G X P can be computed directly from the GIT data [11]. It was shown by Batyrev [14] and by Batyrev et al. [15] that the Laurent polynomial given by assigning the coefficient 1 to the vertices of P, is a mirror partner to X P , with π f P = G X P ; this ansatz for f P is explicitly described by Przyjalkowski [16].
Proposition 1.4. Let P be a smooth Fano polytope. Then f P as given above is a mirror partner to X P .
Proof. The quantum period of X P is computed in the following way. Given a GIT presentation of X P as above, we can write the characters B 1 , . . . , B r as a k × r integer matrix B = B 1 · · · B r called the weight matrix. Here B i is the weight of a torus-invariant divisor D i ⊂ X P . We have that On the other hand, the classical period of f P can be expressed in terms of the matrix A whose columns are the vertices of P (corresponding to the torus-invariant divisors D i ): We see that the sums coincide if we can identify summands via u = vB, but this follows directly from the facts that A and B T are Gale dual matrices and that u is in ker A.

(c) Binomial ansatz
Consider the case when the Fano polytope P ⊂ N Q is reflexive, so that P * ⊂ M Q is also a Fano polytope, but in the lattice M. Every smooth Fano polytope is reflexive. In two dimensions, there are exactly 16 reflexive polygons, and it was observed by Galkin [17] and Przyjalkowski [18] that the Laurent polynomial f P obtained by assigning binomial coefficients to the monomials represented by the lattice points along the edges of P is always a mirror partner to a non-singular del Pezzo surface. Fix an orientation on P, label the vertices a 1 , . . . , a m of P in cyclic order, starting at some arbitrarily chosen vertex a 1 , and set a m+1 := a 1 . For an edge E i = conv{a i , a i+1 } of P, let k i := |E i ∩ N| − 1 denote the lattice length. Define The second sum here removes duplicate contributions from the vertices. Explicit computation gives the following: del Pezzo surfaces). Moreover, eight of the 10 qG-deformation families of non-singular del Pezzo surfaces have a mirror partner arising in this way.
The two missing del Pezzo surfaces here are those of the lowest degree, one and two.
The binomial ansatz is less successful in three dimensions. When applied to the threedimensional reflexive polytopes it generates mirror partners to 92 of the 105 three-dimensional Fano manifolds, but it also generates more than 2000 Laurent polynomials that are not mirror to any three-dimensional Fano manifold. A more successful recipe in three dimensions is the Minkowski ansatz, which we now describe.

(d) Minkowski ansatz
Let P ⊂ N Q be a three-dimensional reflexive polytope. There are 4319 such polytopes [19], and a surprisingly effective partial answer to question 1.1 is given by the Minkowski ansatz ([1], §6). A toric singular point of X P corresponds to a facet F of P, and Altmann [20] tells us that deformation components of that singularity correspond to Minkowski decompositions of F. Altmann's work motivates the following definition. Each facet F of P can be decomposed into irreducible Minkowski summands: We require that each Q i is either a line segment of lattice length one, or is GL(2, Z) Z 2 -equivalent to a triangle A n := conv{0, e 1 , n · e 2 }. If this is the case, we call the decomposition admissible. Laurent polynomials are assigned to the summands in the obvious way: If every facet of P admits an admissible decomposition, the Laurent polynomial f P is given by the 'union' of the f F : Note that the definition of the f Q i guarantees that the coefficients assigned to the codimension >1 faces agree, with binomial coefficients along the edges, so this construction is well defined. Note also that the f F (and hence f P ) depend on the particular choice of admissible decomposition: different decompositions of F can assign different coefficients to the monomials corresponding to F • ∩ N. Example 1.6. Consider the three-dimensional reflexive polytope with seven vertices given by The facets consist of a hexagon F and six A 1 -triangles. (ii) the Minkowski sum of two A 1 -triangles.
The Minkowski ansatz gives the Laurent polynomial where c = 2 in case (i), and c = 3 in case (ii). The two period sequences are: The corresponding Fano manifolds are the hypersurface MM 2-32 of bidegree (1, 1) in P 2 × P 2 , and MM 3-27 = P 1 × P 1 × P 1 . Both Fano manifolds have a qG-degeneration to X P : see, for example, [21,22]. The notation MM k-n here indicates the nth three-dimensional Fano manifold of Picard rank k as classified by Mori & Mukai [23], with the ordering as in [4].

(e) Mutation of Laurent polynomials
], and define the map x v a n y n .
If g := μ(f ), then an application of the change-of-variables formula to the period integral (1.2) gives π f = π g .
Although g is a rational function, in general it need not be a Laurent polynomial. If g is a Laurent polynomial and if f is a mirror partner to X, then g is also a mirror partner for X. Write where all but finitely many of the P i are zero. Then g is a Laurent polynomial if and only if for With this in mind, we make the following definition.
We say that f is mutable with respect to (w, a) if in which case we call g a mutation of f and a a factor.
is a monomial, then μ w,a is simply a monomial change of basis, so we regard the mutation as trivial. Similarly, we regard (w, a) and (w, x v a), x v ∈ C[w ⊥ ∩ N] as defining equivalent mutations, that is, the factor a is considered only up to 'translation'. Typically, if two Laurent polynomials f and g are related via a change of basis then we are inclined to regard them as equivalent; however, we shall see below that it can be important to remember how they were obtained. Example 1.8. Following [24] consider the Laurent polynomial Write: There are three different mutations of f , given by Here all three mutations give equivalent results, up to monomial change of basis. However, it is important to remember that we used three different mutations.

(f) Mutation of polytopes
Let P = Newt(f ) be the Newton polytope of a Laurent polynomial f . We want to be able to work at the level of Newton polytopes-the 'correct' choice of coefficients of the Laurent polynomial f is addressed later. We therefore introduce the following definition, which records the effect of mutation on the Newton polytope of f . Definition 1.9. Let N be a lattice and let w ∈ M be a primitive vector in the dual lattice. Then w induces a grading on N Q . Let A ⊂ w ⊥ := {v ∈ N Q | w(v) = 0} be a convex lattice polytope. Let P ⊂ N Q be a convex lattice polytope and define We say that P is mutable with respect to (w, A) if there exist convex lattice polytopes R i ⊂ w ⊥ (allowing the possibility that If this is the case, then the mutation of P is given by the convex lattice polytope and we call A a factor. Although the existence of a choice of {R i } is required by the definition of mutation, it turns out that the resulting mutation does not depend on the choice made ([2], proposition 1). Hence we can safely omit the {R i } from the notation.
with Newt(f ) = P and Newt(g) = Q, the combinatorial statement (1.5) does not imply the existence of a ∈ C[w ⊥ ∩ N] such that g = μ w,a (f ).
As in the case of Laurent polynomials, if v ∈ w ⊥ ∩ N then P ∼ = μ w,v (P), and more generally μ w,A+v (P) ∼ = μ w,A (P). Thus the factor A is considered only up to translations in w ⊥ ∩ N. Although we typically consider a polytope as being defined up to GL(N)-equivalence, we will see below that it is useful to distinguish between different choices of mutation data.
Given the mirror symmetry perspective that we adopt, a key property of mutation is that it sends Fano polytopes to Fano polytopes:

The mutation graph and rigid Laurent polynomials
In this section, we will introduce rigid maximally mutable Laurent polynomials. This is a class of Laurent polynomials that provides mirror partners to all three-dimensional Fano manifolds, and more generally, we believe, to terminal Fano varieties in every dimension. Roughly speaking, rigid Laurent polynomials are those that are uniquely determined by the mutations that they admit, and maximally mutable Laurent polynomials (MMLPs) are those that admit as many mutations as possible. To make this precise, we first introduce the mutation graph of a Laurent polynomial f .
Identifying mutations of f that differ by GL(N)-equivalence can lead to problems if f admits non-trivial automorphisms-cf. example 1.8. But no non-trivial shear transformation can ever be an automorphism of f , and so it is natural to identify mutations that differ by a shear.
Since w-shears act on N, they act on C(N). Consider a mutation g = μ w,a (f ). If we multiply the factor a by a monomial x b , b ∈ w ⊥ ∩ N, then μ w,ax b (f ) is related to g by a w-shear. Thus, considering a up to multiplication by monomials in C[w ⊥ ∩ N] gives g up to the action of w-shears. We write for the equivalence class. In particular, g ∈ μ w,ax w ⊥ ∩N (f ).
We can now define the mutation graph G f of f . This will be an undirected graph with vertices labelled by GL(N)-equivalence classes of polytopes. Given a primitive vector w ∈ M and a Laurent Here w denotes the linear span of w. Note that a and w determine the pair L(w, a), but that the converse is false.
Before introducing the mutation graph, we fix normalization conditions for the Laurent polynomials that we consider.  The mutation graph G f of f is defined by removing the labels from the edges of G and changing the labels of the vertices from g to the GL(N)-equivalence class of Newt(g).

Definition 2.5.
We partially order the mutation graphs of Laurent polynomials by saying that G f ≺ G g whenever there is a label-preserving injection G f → G g . A Laurent polynomial f is maximally mutable (or for short, f is an MMLP) if Newt(f ) is a Fano polytope, the constant term of f is zero, and G f is maximal with respect to ≺. Definition 2.6. A maximally mutable Laurent polynomial f is rigid if the following holds: for all g such that the constant term of g is zero and Newt(f ) = Newt(g), if G f = G g then f = g. We believe that MMLPs form the correct class of Laurent polynomials to consider when searching for mirrors to Fano varieties. In particular, as we will justify below, we expect that every mirror to a Fano manifold is a rigid MMLP.
We re-emphasize that all of our Laurent polynomials (and mutation factors) here are normalized. We expect (2.1) to characterize rigid MMLPs in all dimensions.

Maximal mutability in dimension 2
Mutation and maximally mutable Laurent polynomials are particularly well behaved in two dimensions, as we will now explain. The results in this section complete an important part of the programme described in [6], which recasts the classification of del Pezzo surfaces with orbifold singularities in terms of mirror symmetry.

(a) Singularity content
Let us consider mutations of polytopes in dimension two. Each edge of a Fano polygon P ⊂ N Q determines a point, which may be singular, in the toric surface X P . We first describe the effect of mutation on these singularities: see [7,20,27]. An edge e of a Fano polygon P determines the cone σ over e, and hence a torus-fixed point in the toric variety X P . A neighbourhood of this point is the affine toric variety X σ determined by σ , that is, the cyclic quotient singularity (1/b)(1, c − 1), where the rays {u, v} of σ are such that u, v and ((c − 1)/b)u + (1/b)v generate N. Write b = dr and c = dk, where d = gcd{b, c}. Then X σ is the cyclic quotient singularity (1/dr)(1, kd − 1). Furthermore, r is the lattice height of e above the origin, and d is the lattice length of e.
When d = nr for some n ∈ Z, the singularity X σ is called a T-singularity and σ is called a T-cone. If n = 1 then X σ is a primitive T-singularity and σ is a primitive T-cone. T-singularities are  qG-smoothable [28]. A general T-singularity (1/nr 2 )(1, knr − 1) admits a crepant partial resolution into n primitive T-singularities (1/r 2 )(1, kr − 1); this amounts to the fact that, since d = nr, the cone σ can be decomposed as a union of n primitive T-cones (figure 2).
When d = s with 0 < s < r, the singularity X σ is called an R-singularity and σ is called an R-cone. R-singularities are qG-rigid. When d = nr + s for some non-negative n ∈ Z and 0 < s < r, the singularity X σ = (1/dr)(1, kd − 1) admits a (non-unique) crepant partial resolution into n primitive T-singularities and the single R-singularity (1/sr)(1, ks − 1); this corresponds to the (non-unique) decomposition of σ into n primitive T-cones and one R-cone (figure 3).
The singularity X σ qG-deforms to the R-singularity (1/sr)(1, ks − 1). We define the residue of σ to be and the singularity content of σ to be the pair (n, res(σ )).

Definition 3.2.
Let P be a Fano polygon with edges e 1 , . . . , e m , and suppose that the singularity content of the cone over e i is (n i , S i ). Let n = n 1 + · · · + n m , and let B be the multiset containing those S i such that S i = ∅, counted with multiplicity. The singularity content of P is SC(P) = (n, B).

Theorem 3.4 ([7], theorem 3.8). Singularity content is a mutation invariant.
If P is a Fano polygon with singularity content (n, B) then a generic qG-deformation of X P has singularities given by B. Furthermore there is a (non-unique) toric crepant partial resolution of Y → X P such that the singularities of Y are given by n primitive T-singularities together with B.  Let C be an R-cone with primitive ray generators ρ 1 , ρ 2 ∈ N. Let u ∈ M be the primitive normal vector to the edge conv{ρ 1 , ρ 2 } that is positive on C, and let r = u(ρ 1 ) = u(ρ 2 ). The set R C of residual points of C is Definition 3.6. Let P ⊂ N Q be a Fano polygon with singularity content (n, B), and let k = |B|. We say that R ⊂ N is a choice of residual points for P if and only if there exists a crepant subdivision of the spanning fan for P into n T-cones and k R-cones C 1 , . . . , C k such that Any two choices R 1 , R 2 of residual points for P satisfy |R 1 | = |R 2 |.

(b) MMLPs
We next prove that, in dimension two, mutations of MMLPs are in one-to-one correspondence with mutations of the underlying Newton polygons. A mutation of a polygon P amounts to removing a number of primitive T-cones from one edge of P and inserting the same number of primitive T-cones on the opposite side: see figure 4.
( ‡) To prove the Proposition, we need to prove statement ( ‡) for all h ≥ 0. It holds trivially for h = 0.
(ii) Base case: h = 1 Points of P-height 1 lie on the boundary of P. Consider an edge e of P. Without loss of generality we may assume that e is horizontal and at positive vertical height r above the origin. Up to overall multiplication by a monomial, the coefficients of f along e are where d = n e r + s and 0 ≤ s < r, for some b 1 , . . . , b d−1 . The mutability condition gives that

(iii) Induction step
Suppose that H satisfies 0 < H < 1 and that statement ( ‡) holds for all h > H. We will prove ( ‡) for h = H. Let v ∈ P be a lattice point of P-height H. Since P is Fano, v lies in the cone C e over a unique edge e. Without loss of generality, we may assume that e is horizontal and at positive vertical height r above the origin. Let d denote the lattice length of e, and write d = n e r + s with 0 ≤ s < r. Consider the horizontal line L through v; this is at vertical height Hr above the origin. Lattice points on L fall into three classes: residual points in C e , non-residual points in C e , and points outside C e . Convexity implies that the points on L outside C e are at P-height greater than H: see figure 5. By the induction hypothesis, therefore, coefficients of these points are fixed as affine-linear combinations of coefficients of residual points of height greater than H. for some coefficients c i . A primitive T-cone with lattice height r contains exactly h lattice points at lattice height h, if 0 < h < r, and so if L ∩ P contains no residual points and no points outside C e then m < rHn e . In this case (3.3) gives that b 0 = · · · = b m = 0, and statement ( ‡) holds. Otherwise (3.3) holds with s ≥ 0 and s + 1 equal to the total number of points in L ∩ P that are either residual or lie outside C e . Statement ( ‡) for h = H follows if we can solve (3.3) uniquely for c i in terms of the coefficients b k 0 , . . . , b k s of points on L that are either residual or lie outside C e . For this, we use the same matrix-invertibility argument as above. This completes the induction step, and the proof of proposition 3.7.

Proposition 3.9. Let F be a Laurent polynomial in two variables with Fano Newton polygon. Then F is maximally mutable if and only if it is a specialization of the general Laurent polynomial f obtained from proposition 3.7 applied to P = Newt(F).
Proof. Suppose that P is a Fano polytope, and that f is a general Laurent polynomial with zero constant term and Newton polytope P that satisfies the mutability conditions in proposition 3.7. These conditions imply that the mutation graph of f is n-valent at the vertex defined by f , where (n, B) is the singularity content of P. This is the maximum possible valency at that vertex, because P admits only n non-trivial mutations. Suppose now that f is mutable with respect to (w, a), and that the Newton polytope of the mutation μ w,a (f ) is Q. Let g be the general Laurent polynomial with zero constant term and Newton polytope Q provided by proposition 3.7. Fix residual points R P for P and R Q for Q, noting that, because singularity content is a mutation invariant, |R P | = |R Q |. We have where the coefficients c v are affine-linear functions of {c w | w ∈ R P } and the coefficients c v are affine-linear functions of {c w | w ∈ R Q }. The coefficients of the mutation μ w,a (f ) are affine-linear functions of {c v | v ∈ P ∩ N}, and hence of { c w | w ∈ R P }. Taking the coefficients of μ w,a (f ) at elements of R Q gives an affine-linear isomorphism between {c w | w ∈ R P } and {c w | w ∈ R Q }, and under this isomorphism μ w,a (f ) coincides with g. In particular, the mutation graph of f is also n-valent at the vertex defined by g. Thus, the mutation graph of f has maximal valency at each vertex, and therefore f is maximally mutable. Furthermore, G F ≺ G f by construction. Since F is maximally mutable it follows that F is obtained from f by specialization of coefficients.
This also proved:   Figures 6 and 7 compare the mutation graph of the rigid MMLP f = y + ((1 + x) 2 /xy) and the quiver mutation graph [25] for Newt(f ). A polytope at a vertex denotes the GL(2, Z)-equivalence class of that polytope.
As discussed above, these results complete an important part of the program described in [6]. Fano polygons with singularity content (n, ∅) for some n fall into exactly 10 mutationequivalence classes ( [25], theorem 1.2). Each mutation class supports exactly one mutation class of rigid MMLP, provided by theorem 3.9, and these correspond one-to-one with qG-deformation families of smooth del Pezzo surfaces. Under this correspondence, the classical period π f of a rigid MMLP f matches with the regularized quantum period G X of the del Pezzo surface (   Combining [30] with [31] proves the analogous result for del Pezzo surfaces with 1 3 (1, 1) singularities: see [6].
Tveiten [32] has studied the geometry of MMLPs in two dimensions. A Laurent polynomial f in two variables determines a pencil of curves Y f → P 1 as follows. Both f and the unit monomial determine sections of the anti-canonical divisor on Y P , where P = Newt(f ) and Y P is the toric variety defined by the normal fan to P. Resolving singularities of Y P and resolving basepoints of the rational map Y P P 1 defined by [1 : f ] defines the pencil Y f → P 1 . Writing p for the map Y f → P 1 , we can consider the monodromy about 0 ∈ P 1 of the local system R 1 p ! Q. Equivalently, this is the monodromy about zero of the Picard-Fuchs differential operator L f that annihilates the classical period π f . Theorem 3.14 ([32], theorem 4.17). Let f be a maximally mutable Laurent polynomial in two variables, let P be its Newton polygon, and let π f be its classical period. The monodromy about zero of L f determines and is determined by the singularity content of P.
We have seen that singularity content is a mutation invariant of Fano polygons. It is not, however, a complete invariant. Example 3.15. Consider the polygons P and Q given, respectively, by the convex hull of the rays of a fan for P 1 × P 1 and for the Hirzebruch surface F 1 . Both P and Q have singularity content (4, ∅), but they are not mutation-equivalent. Proposition 3.9 determines unique MMLPs f with Newton polytope P and g with Newton polytope Q, and these MMLPs have distinct classical periods: π f (t) = 1 + 4t 2 + 36t 4 + 400t 6 + · · · and π g (t) = 1 + 2t 2 + 6t 3 + 6t 4 + 60t 5 + 110t 6 + · · · .
We conjecture that the classical period of the general MMLP given by proposition 3.9 is a complete invariant for mutation of Fano polygons: cf. ( [6], conjectures A and B).

Some three-dimensional results
The Minkowski ansatz [1] is extremely successful at recovering mirrors for the 98 threedimensional Fano manifolds with very ample anti-canonical bundle, but it has several drawbacks.
(i) The ansatz can only be applied to reflexive polytopes, and so cannot be used to recover the seven Fano manifolds that do not have very ample anti-canonical bundle (although mirrors for these cases are known: see [4], Proof. The rigid MMLPs supported on each of the 4319 three-dimensional reflexive polytopes were computed using the computer algebra system MAGMA [34]. These include every Minkowski polynomial mirror to one of the 98 three-dimensional Fano manifolds with very ample anticanonical bundle. The rigid MMLPs are listed in the supplementary material; they fall into 98 mutation-equivalence classes, which correspond one-to-one to the three-dimensional Fano manifolds just discussed. Under this correspondence, the classical period π f of a rigid MMLP f matches with the regularized quantum period G X of the corresponding Fano manifold [4]. Mirrors for the remaining seven Fano manifolds can be constructed from the descriptions in [4]. In each case, it was verified that the resulting Laurent polynomial is rigid.
One can try to produce mirrors supported on a reflexive polytope that are not rigid MMLPs. There are two obvious approaches.
(i) Start with a known Laurent polynomial mirror to a Fano manifold and repeatedly mutate, looking for new mirrors supported on a reflexive polytope. (ii) Start with the quantum period sequence for a Fano manifold and assign coefficients to the lattice points in a reflexive polytope in order to recover a Laurent polynomial with the correct period sequence.
We systematically applied both approaches, but were unable to produce any non-rigid mirrors.
Note that a three-dimensional analogue of theorem 3.12 cannot hold: when one considers nonreflexive Newton polytopes there exist rigid MMLPs that are not mirror to a Fano manifold.  [35,36]. Not all such potential Hilbert series are realized by genuine threefolds, but this one is: it arises from a complete intersection X 3,3 ⊂ P(1 5 , 2). The variety X 3,3 has a terminal singularity of type 1 2 (1, 1, 1). A Laurent polynomial mirror to X 3,3 , computed via [11,37], is f = z(1 + x + y) 3 This is readily seen to be a rigid MMLP. This supports a unique rigid MMLP given by By applying Laurent inversion [30], we see that this defines a hypersurface of type (2,2) in the four-dimensional Fano toric variety with weight data 1 0 1 −1 2 1 0 1 1 1 0 0 .
Thus f is a mirror to a Fano threefold that realizes the Hilbert series with ID 35296 in [36]; this variety has codimension 10 in P(1 12 , 2 2 ) and has two terminal singularities of type 1 2 (1, 1, 1). Examples 4.2 and 4.3 together illustrate two points that we will explore more fully elsewhere: that rigid MMLPs give a new tool [40] for exploring the (unknown) classification of terminal Fano threefolds [35], and that they can help to find simple constructions of singular Fano threefolds that are conjectured to exist but which, as they have high codimension, otherwise seem complicated and hard to reach [41]. They also illustrate that, although proving that a given Laurent polynomial f is maximally mutable is hard, proving that f is rigid maximally mutable is often much easier. Indeed any normalized Laurent polynomial f with zero constant term and Fano Newton polytope such that the coefficients of f are determined by the mutations that f supports is automatically rigid maximally mutable.

Remarks on higher dimension
Every known mirror to a Fano manifold in dimension four or more is a rigid MMLP [5,[42][43][44]. But when considering mirror symmetry for higher-dimensional Fano varieties, one cannot just restrict attention to smooth manifolds. For example, any simplicial terminal Fano polytope P will support a unique normalized Laurent polynomial-which is a rigid MMLP-with X P a Q-factorial Fano variety with at worst terminal singularities. Such varieties are rigid under deformation [45]. So even the foundational ansatz for mirror symmetry for toric varieties, the Givental/Hori-Vafa mirror [11,37], leads us in higher dimensions to varieties with terminal singularities. This feature of Mirror Symmetry was masked in dimensions up to three, where the focus so far has been on Gorenstein Fano varieties: in dimensions up to three, Gorenstein terminal varieties with quotient singularities are smooth. In dimensions four and higher, terminal singularities are unavoidable [46].
Terminal singularities are, of course, a very natural class of singularities. Introduced by Reid, they are required by the Minimal Model Program [47,48], and the classification of terminal Fano varieties is a fundamental open problem in birational geometry. The following conjecture suggests that one approach to this classification is via rigid MMLPs; the ideas involved arose from many conversations with Alessio Corti and the rest of our collaborators in the Fanosearch project. Recall that a Fano variety X is of class TG if it admits 3 a qG-degeneration with reduced fibres to a normal toric variety [6].
Conjecture 5.1. Rigid MMLPs in n variables (up to mutation) are in one-to-one correspondence with pairs (X, D), where X is a Fano n-fold of class TG with terminal locally toric qG-rigid singularities and D ∈ |−K X | is a general elephant (up to qG-deformation).
Data accessibility. The data are contained in the electronic supplementary material. Authors' contributions. The authors contributed equally to the research presented in this paper. Competing interests. We declare we have no competing interests. Funding. A.K. is supported by EPSRC Fellowship EP/N022513/1. T.C. and G.P. are funded by ERC Consolidator grant no. 682603 and EPSRC Programme grant no. EP/N03189X/1.