Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
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Maximally mutable Laurent polynomials

Tom Coates

Tom Coates

Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK

[email protected]

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,
Alexander M. Kasprzyk

Alexander M. Kasprzyk

School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK

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,
Giuseppe Pitton

Giuseppe Pitton

Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK

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and
Ketil Tveiten

Ketil Tveiten

Department of Mathematics, Uppsala University, PO Box 256, Uppsala 751 05, Sweden

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    Abstract

    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.

    1. Introduction

    Recently, a new approach to the classification of Fano manifolds was proposed [1]. This centres on mirror symmetry, in the form of a conjectural relationship between Fano manifolds and Laurent polynomials. An n-dimensional Fano manifold X determines the regularized quantum period

    G^X(t)=1+k=2k!cktk. 1.1
    This is a generating function for certain Gromov–Witten invariants of X: roughly speaking, the coefficient ck 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 fC[x1±1,,xn±1], and when this happens there is a close connection between the geometry of X and of f. More precisely, given a Laurent polynomial fC[x1±1,,xn±1], one can form its classical period:
    πf(t)=(12πi)n|x1|==|xn|=111tfdx1x1dxnxn,tC,|t|. 1.2
    The Taylor expansion of πf, which we also denote πf, can be readily calculated [2,3]:
    πf(t)=k=0coeff1(fk)tk. 1.3
    Here coeff1(fk) denotes the coefficient of the constant term of fk. If
    G^X=πf,
    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 Zn; 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 XP associated with the Newton polytope P=Newt(f) of f [6,7]. Here XP 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 XP and XQ, 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 PNQ:=NZQ be a convex lattice polytope.

    Question 1.1.

    Does there exist a Laurent polynomial

    f=vPNcvxv,
    with coefficients cv 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:=PP of P: if 0P then, for any Laurent polynomial f with Newt(f)=P, the period πf will be constant, and hence cannot be the quantum period of a Fano manifold; if 0P then we can reduce to a lower-dimensional situation by considering the smallest dimensional face F of P with 0F.

    (ii)

    We can assume that dim(P)=n, since otherwise we can restrict to the sublattice given by spanQ(P)N.

    (iii)

    Consider the sublattice

    N:=vPNvZ,
    of N generated by the lattice points of P. Restricting to the sublattice N will not change the corresponding period πf, since coeff1(fk) 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 XP 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 XP is an n-dimensional toric Fano variety.

    Definition 1.1.

    A convex lattice polytope PNQ 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 XP, 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 XP, 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.

    Example 1.3.

    Consider the three-dimensional Fano polytopes

    P:=conv{(1,0,0),(0,1,0),(1,1,0),(1,2,1),(1,2,1)}
    and
    Q:=conv{(1,0,0),(0,1,0),(1,1,0),(1,2,3),(1,2,3)}.

    We have that |PN|=|QN|=6, with the only non-vertex lattice point in each case corresponding to the origin. The points of Q generate the sublattice e1Z+e2Z+3e3Z of index three, and restricting Q to this sublattice gives P. The Laurent polynomials

    f:=x+y+1xy+xy2z+1xy2zandg:=x+y+1xy+xy2z3+1xy2z3,
    which have Newton polytopes P and Q, respectively, generate the same period sequence:
    πf(t)=πg(t)=k=0m=0(2k+3mk,k,m,m,m)t2k+3m=1+2t2+6t3+6t4+120t5+110t6+1260t7+.
    This agrees with the period sequence for P1×P2 given by Givental [11], and in fact the toric variety XP is equal to P1×P2. The anti-canonical degree of XP is (KXP)3=Vol(P)=54, where
    P:={uMQu(v)1for allvP}
    is the dual (or polar) polytope to P in the dual lattice M:=Hom(N,Z), and Vol() denotes the lattice-normalized volume. The degree of XQ differs from that of XP by a factor of three, that being the index of the sublattice, so that (KXQ)3=Vol(Q)=1/3Vol(P)=18. In particular, even though πf=πg=G^P1×P2, XQ cannot be a qG-degeneration of P1×P2.

    (b) Vertex ansatz

    Let PNQ be a smooth Fano polytope. That is, for each facet F of P, the vertices of F generate the lattice N. In two dimensions, there are five smooth Fano polygons; in three dimensions, there are only 18 smooth Fano polytopes [12,13]. Since P is a smooth Fano polytope, the corresponding toric variety XP is a Fano manifold. Any toric Fano variety can be expressed as a GIT quotient of the form Cr//(C×)k—this generalizes the construction of projective space Pr1 as (Cr{0})/C×=Cr//C×, and amounts to specifying1 the characters B1,,Br of (C×)k that define the action of (C×)k on Cr. The quantum period G^XP 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

    fP:=vvert(P)xv,
    given by assigning the coefficient 1 to the vertices of P, is a mirror partner to XP, with πfP=G^XP; this ansatz for fP is explicitly described by Przyjalkowski [16].

    Proposition 1.4.

    Let P be a smooth Fano polytope. Then fP as given above is a mirror partner to XP.

    Proof.

    The quantum period of XP is computed in the following way. Given a GIT presentation of XP as above, we can write the characters B1,,Br as a k×r integer matrix

    B=(B1Br)
    called the weight matrix. Here Bi is the weight of a torus-invariant divisor DiXP. We have that
    G^XP(t)=v:v,Bi0i(iv,Bi)!iv,Bi!tiv,Bi.
    On the other hand, the classical period of fP can be expressed in terms of the matrix A whose columns are the vertices of P (corresponding to the torus-invariant divisors Di):
    πf(t)=ukerA:ui0i(iui)!iui!tiui.
    We see that the sums coincide if we can identify summands via u=vB, but this follows directly from the facts that A and BT are Gale dual matrices and that u is in kerA.

    (c) Binomial ansatz

    Consider the case when the Fano polytope PNQ is reflexive, so that PMQ 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 fP 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 a1,,am of P in cyclic order, starting at some arbitrarily chosen vertex a1, and set am+1:=a1. For an edge Ei=conv{ai,ai+1} of P, let ki:=|EiN|1 denote the lattice length. Define

    fP:=i=1mxai(1+xwi)kii=1mxai,wherewi:=1ki(ai+1ai)Nis primitive. 1.4
    The second sum here removes duplicate contributions from the vertices. Explicit computation gives the following:

    Proposition 1.5.

    Let PNQ be a reflexive polygon. The Laurent polynomial fP defined by (1.4) is a mirror partner to a non-singular del Pezzo surface (or rather, to a qG-deformation family of non-singular 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 three-dimensional 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 PNQ 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 XP 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:

    F=Q1++Qr,where dim(Qi)1.
    We require that each Qi is either a line segment of lattice length one, or is GL(2,Z)Z2-equivalent to a triangle An:=conv{0,e1,ne2}. If this is the case, we call the decomposition admissible. Laurent polynomials are assigned to the summands in the obvious way: if Qi=conv{a0,a0+a1}, where a0,a1N, a1 primitive, then fQi=xa0(1+xa1); if Qi=conv{a0,a0+a1,a0+na2}, where a0,a1,a2N, a1 and a2 primitive, then fQi=xa0(xa1+(1+xa2)n). Define
    fF:=i=1rfQi.
    If every facet of P admits an admissible decomposition, the Laurent polynomial fP is given by the ‘union’ of the fF:
    fP:=vPNcvxv,wherecvis the coefficient ofxvin one of thefF.
    Note that the definition of the fQi 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 fF (and hence fP) depend on the particular choice of admissible decomposition: different decompositions of F can assign different coefficients to the monomials correspondingto FN.

    Example 1.6.

    Consider the three-dimensional reflexive polytope with seven vertices given by

    P:=conv{(0,1,1),(1,1,1),(1,0,1),(0,1,1),(1,1,1),(1,0,1),(0,0,1)}NQ.
    The facets consist of a hexagon F and six A1-triangles.

    Inline Graphic

    There are two admissible decompositions of F:

    (i)

    the Minkowski sum of three line segments of lattice length one;

    Display Formula

    (ii)

    the Minkowski sum of two A1-triangles.

    Display Formula

    The Minkowski ansatz gives the Laurent polynomial

    fc:=yz+xyz+xz+1yz+1xyz+1xz+cz+z,
    where c=2 in case (i), and c=3 in case (ii). The two period sequences are:
    πf2(t)=1+4t2+60t4+1120t6+andπf3(t)=1+6t2+90t4+1860t6+
    The corresponding Fano manifolds are the hypersurface MM2--32 of bidegree (1,1) in P2×P2, and MM3--27=P1×P1×P1. Both Fano manifolds have a qG-degeneration to XP: see, for example, [21,22]. The notation MMk--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

    Let fC[x±1,y±1], aC[x±1], and define the map

    μ:C(x,y)C(x,y),xvynxvanyn.
    If g:=μ(f), then an application of the change-of-variables formula to the period integral (1.2) gives
    πf=πg.
    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
    f=iZPi(x)yifor somePiC[x±1],
    where all but finitely many of the Pi are zero. Then g is a Laurent polynomial if and only if for each iZ<0, there exists riC[x±1] such that Pi=ria|i|; in this case
    g=μ(f)=iZ<0riyi+jZ0PjajyjC[x±1,y±1].
    With this in mind, we make the following definition.

    Definition 1.7.

    Let N be a lattice and let wM be a primitive vector in the dual lattice. Then w induces a grading on C[N]. Let aC[wN] be a Laurent polynomial in the zeroth piece of C[N], where wN={vNw(v)=0}. The pair (w,a) defines an automorphism of C(N) given by

    μw,a:C(N)C(N),xvxvaw(v).
    Let fC[N]. We say that f is mutable with respect to (w,a) if
    g:=μw,a(f)C[N],
    in which case we call g a mutation of f and a a factor.

    If a=xvC[wN] is a monomial, then μw,a is simply a monomial change of basis, so we regard the mutation as trivial. Similarly, we regard (w,a) and (w,xva), xvC[wN] 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 f=x+y+1/(xy)C[x±1,y±1]. Write:

    x+y+1xy=x(1+1x2y)+y=1xy(1+xy2)+x=x(1+yx)+1xy.
    There are three different mutations of f, given by

    Inline Graphic

    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 wM be a primitive vector in the dual lattice. Then w induces a grading on NQ. Let Aw:={vNQw(v)=0} be a convex lattice polytope. Let PNQ be a convex lattice polytope and define

    Pi:=conv{aPNw(a)=i},noting that often Pi=.
    We say that P is mutable with respect to (w,A) if there exist convex lattice polytopes Riw (allowing the possibility that Ri=) such that
    {vvert(P)w(v)=i}Ri+|i|APifor each iZ<0.
    If this is the case, then the mutation of P is given by the convex lattice polytope
    μw,A(P):=conv(iZ<0RijZ0(Pj+jA)),
    and we call A a factor.

    Although the existence of a choice of {Ri} 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 {Ri} from the notation.

    Remark 1.10.

    Suppose that g=μw,a(f), for some f, gC[N], aC[wN]. Then

    Newt(g)=μw,Newt(a)(Newt(f)).
    Conversely, if
    Q=μw,A(P) 1.5
    then there exists some choice of f, gC[N] and aC[wN] with Newt(f)=P, Newt(g)=Q, and Newt(a)=A, such that g=μw,a(f). Note however that for fixed f, gC[N] with Newt(f)=P and Newt(g)=Q, the combinatorial statement (1.5) does not imply the existence of aC[wN] such that g=μw,a(f).

    As in the case of Laurent polynomials, if vwN then Pμw,v(P), and more generally μw,A+v(P)μw,A(P). Thus the factor A is considered only up to translations in wN. 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:

    Proposition 1.11 ([2], proposition 2).

    Let Q=μw,A(P) be a mutation of a convex lattice polytope PNQ. Then P is a Fano polytope if and only if Q is a Fano polytope.

    2. 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.

    Definition 2.1.

    A transformation BSL(N) is called a w-shear, where wM, if B|w=Id.

    Since w-shears act on N, they act on C(N). Consider a mutation g=μw,a(f). If we multiply the factor a by a monomial xb, bwN, then μw,axb(f) is related to g by a w-shear. Thus, considering a up to multiplication by monomials in C[wN] gives g up to the action of w-shears. We write

    μw,axwN(f)
    for the equivalence class. In particular, gμw,axwN(f).

    We can now define the mutation graph Gf of f. This will be an undirected graph with vertices labelled by GL(N)-equivalence classes of polytopes. Given a primitive vector wM and a Laurent polynomial aC[wN], write

    L(w,a):=(w,axwN).
    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.

    Definition 2.2.

    A Laurent polynomial fC[N] is normalized if for all vertices v of Newt(f), the coefficient of the monomial xv in f is 1.

    Convention 2.3.

    From here onwards, we assume that all Laurent polynomials (and all mutation factors) are normalized. Similarly, although our Laurent polynomials are defined over C, our expectation is that Laurent polynomials that are mirror to Fano manifolds have coefficients that are non-negative integers. From here onwards, we will require that all Laurent polynomials (and all mutation factors) have non-negative integer coefficients. Recall our standing convention that if fC[N], then the exponents of monomials in f generate N.

    Definition 2.4.

    Given a Laurent polynomial f, consider the graph G with vertex labels that are Laurent polynomials and edge labels that are pairs L(w,a), defined as follows. Write (v) for the label of a vertex vV(G), and (e) for the label of an edge eE(G).

    (i)

    Begin with a vertex labelled by the Laurent polynomial f.

    (ii)

    Given a vertex v, set g:=(v). For each (w,a), dega>0, such that g is mutable with respect to (w,a) and either:

    (a)

    there does not exist an edge with endpoint v and label L(w,a); or

    (b)

    for every edge e=vv¯ with (e)=L(w,a) we have that

    (v)μw,axwN(g);

    pick a representative gμw,axwN(g) and add a new vertex v and edge vv¯ labelled by g and L(w,a), respectively.

    The mutation graph Gf 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 GfGg whenever there is a label-preserving injection GfGg. 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 Gf 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 Gf=Gg then f=g.

    Remark 2.7.

    Mutations leave the constant term of a Laurent polynomial unchanged; furthermore, the choice of constant term of f does not affect any other coefficient in a mutation of f. We have chosen to fix the constant term as zero in definition 2.5 because the regularized quantum period (1.1) of a Fano manifold X always satisfies c1=0. Similarly, our choice of normalization conditions (definition 2.2) reflects the properties expected of Laurent polynomial mirrors to Fano manifolds.

    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.

    In dimension two, the close relationship between mutations of Fano polygons and mutations of quivers2 means that rigidity of f can be detected ‘locally’ in the mutation graph. That is, one can determine whether f is rigid by considering only mutations of f itself. Define

    Sf:={(w,a)fis mutable with respect to(w,a)}
    and, given any set S of pairs (w,a) with wM primitive and aC[wN], define
    LP(S):={fC[N]|Newt(f)=Pandfis mutable withrespect to(w,a)for all(w,a)S}.
    Then f is a rigid MMLP if and only if
    LNewt(f)(Sf)={f}. 2.1
    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.

    3. 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 PNQ determines a point, which may be singular, in the toric surface XP. We first describe the effect of mutation on these singularities: see [7,20,27].

    Example 3.1.

    Consider the mutation of Fano polygons depicted in figure 1, where w=(0,1)M and A=conv{(0,0),(1,0)}w. This mutation has the effect of removing a line segment of length three at height 3, and introducing a line segment of length two at height 2.

    Figure 1.

    Figure 1. A mutation of Fano polygons.

    An edge e of a Fano polygon P determines the cone σ over e, and hence a torus-fixed point in the toric variety XP. A neighbourhood of this point is the affine toric variety Xσ determined by σ, that is, the cyclic quotient singularity (1/b)(1,c1), where the rays {u,v} of σ are such that u, v and ((c1)/b)u+(1/b)v generate N. Write b=dr and c=dk, where d=gcd{b,c}. Then Xσ is the cyclic quotient singularity (1/dr)(1,kd1). Furthermore, r is the lattice height of e above the origin, and d is the lattice length of e.

    When d=nr for some nZ, the singularity Xσ is called a T-singularity and σ is called a T-cone. If n=1 then Xσ is a primitive T-singularity and σ is a primitive T-cone. T-singularities are qG-smoothable [28]. A general T-singularity (1/nr2)(1,knr1) admits a crepant partial resolution into n primitive T-singularities (1/r2)(1,kr1); 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).

    Figure 2.

    Figure 2. A decomposition of the T-cone (1/322)(1,5) into three primitive T-cones (1/22)(1,1).

    When d=s with 0<s<r, the singularity Xσ is called an R-singularity and σ is called anR-cone. R-singularities are qG-rigid. When d=nr+s for some non-negative nZ and 0<s<r, the singularity Xσ=(1/dr)(1,kd1) admits a (non-unique) crepant partial resolution into n primitive T-singularities and the single R-singularity (1/sr)(1,ks1); this corresponds to the (non-unique) decomposition of σ into n primitive T-cones and one R-cone (figure 3).

    Figure 3.

    Figure 3. A decomposition of the cone 1/(73)(1,13) into two primitive T-cones (1/32)(1,5) and one R-cone (1/3)(1,1).

    The singularity Xσ qG-deforms to the R-singularity (1/sr)(1,ks1). We define the residue of σ to be

    res(σ)={if s=01sr(1,ks1)otherwise,
    and the singularity content of σ to be the pair (n,res(σ)).

    Definition 3.2.

    Let P be a Fano polygon with edges e1,,em, and suppose that the singularity content of the cone over ei is (ni,Si). Let n=n1++nm, and let B be the multiset containing those Si such that Si, counted with multiplicity. The singularity content of P is

    SC(P)=(n,B).

    Example 3.3.

    Returning to figure 1, we see that the mutation changes the singularity content of the cone over the bottom edge from (1,13(1,1)) to (0,13(1,1)); it also introduces a new edge, at the top of the polygon, with singularity content (1,). All other singularities are unchanged. The singularity content of both polygons is (4,{2×13(1,1)}).

    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 XP has singularities given by B. Furthermore there is a (non-unique) toric crepant partial resolution of YXP such that the singularities of Y are given by n primitive T-singularities together with B.

    Definition 3.5.

    Let C be an R-cone with primitive ray generators ρ1, ρ2N. Let uM be the primitive normal vector to the edge conv{ρ1,ρ2} that is positive on C, and let r=u(ρ1)=u(ρ2). The set RC of residual points of C is

    RC:=C{pNu(p)r}{0,ρ1,ρ2}.

    Definition 3.6.

    Let PNQ be a Fano polygon with singularity content (n,B), and let k=|B|. We say that RN 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 C1,,Ck such that

    R=RC1RCk.

    Any two choices R1, R2 of residual points for P satisfy |R1|=|R2|.

    (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.

    Figure 4.

    Figure 4. Mutation of a Fano polygon removes and inserts primitive T-cones.

    Our first result produces, for each Fano polygon P, a non-empty family of Laurent polynomials that is compatible with all mutations of P. Later we will show that these Laurent polynomials are in fact maximally mutable.

    Proposition 3.7.

    Let P be a Fano polygon. For each edge e of P, fix a primitive vector ve in the direction of e and write we for the primitive inward-pointing normal vector to e. Write (ne,Se) for the singularity content of the cone over e, and set ae=(1+xve)ne. There exists a Laurent polynomial f with Newton polytope P and zero constant term that is mutable with respect to (we,ae) for each edge e. Furthermore, the free parameters in the coefficients of the general such f are in bijection with any choice of residual points of P, and are affine-linear functions of the coefficients of such points.

    Before proving proposition 3.7, we give an example that illustrates the method of proof.

    Example 3.8.

    Consider the square P with vertices (2,3), (2,3), (2,1) and (2,1). This has singularity content (9,{13(1,1)}). A normalized Laurent polynomial f with Newton polygon P that satisfies the mutability conditions in proposition 3.7 must have binomial coefficients along the bottom edge: writing

    f=i=13Pi(x)yi
    we have that P1 is divisible by (1+x)4 and is normalized, so P1=x2(1+x)4. Furthermore, considering the top edge, we see that P3 is normalized and is divisible by (1+x)3, so P3=x2(1+x)4. Arguing similarly, the other two edges also carry binomial coefficients.
    Display Formula
    Now consider P2(x)=x2(4+ax+bx2+cx3+4x4). This is divisible by (1+x)2, which forces
    4+ax+bx2+cx3+4x4=(1+x)2(α1+α2x+α3x2).
    We must have α1=α3=4, and a=c=8+α, b=8+2α for some unknown α=α2. Similarly, P1(x)=x2(6+dx+ex2+gx3+6x4), and divisibility by (1+x) forces d=6+β, e=β+γ, g=6+γ for some β and γ.

    Now consider the left-hand edge, and the mutation with w=(1,0). Writing

    f=i=22P¯i(y)xi,
    we see that P¯1(y)=y1(4+hy+dy2+ay3+4y4) is divisible by (1+y)2. This forces
    4+hy+dy2+ay3+4y4=(1+y)2(δ1+δ2y+δ3y2),
    and therefore δ1=δ3=4, h=a=8+δ, d=8+2δ, where δ=δ2. The relation a=8+α=8+δ allows us to eliminate δ, and the relation d=6+β=8+2δ allows us to eliminate β. A similar argument for the right-most edge gives us relations from c and g; we conclude that f has coefficients as follows:
    Display Formula
    The number of free parameters here (one) is equal to the number of residual points in P.

    (i) Proof of proposition 3.7.

    Let us define the P-height of a lattice point pP to be the non-negative rational number r such that p lies on the boundary of rP. The set of P-heights of points in PN is a finite subset of [0,1]. Fix a set of residual points R for P, and write Rh for the set of points in R of P-height at least h. Set f=vPNavxv. We will prove the following statement by descending induction on h:

    ()The coefficients av such that v has P-height h are affine-linear functions of thecoefficients aw for wRh, and these coefficients aw are independent. 3.1
    To prove the Proposition, we need to prove statement () for all h0. It holds trivially forh=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

    1+b1x++bd1xd1+xd, 3.2
    where d=ner+s and 0s<r, for some b1,,bd1. The mutability condition gives that
    1+b1x++bd1xd1+xd=(1+x)rnei=0scixi, 3.3
    for some coefficients ci. This forces c0=cs=1. If s=0 or s=1 then there are no residual points on e, and () holds as (3.1) coincides with (1+x)d. If s>1 then to prove () it suffices to show that we can solve (3.2) uniquely for ci in terms of the coefficients bk1,,bks1 of residual points on e. That is, it suffices to show that the (s1)×(s1) matrix with (i,j) entry
    (nerkij)
    is invertible. But the determinant of this matrix is positive—indeed it counts certain semi-standard Young tableaux, as can be seen by specializing to 1 all variables in the Giambelli formula ([29], eqn A6) that expresses Schur polynomials in terms of elementary symmetric polynomials. Thus statement () holds for h=1.

    (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 vP be a lattice point of P-height H. Since P is Fano, v lies in the cone Ce 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=ner+s with 0s<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 Ce, non-residual points in Ce, and points outside Ce. Convexity implies that the points on L outside Ce 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.

    Figure 5.

    Figure 5. Lattice points on L outside the cone Ce are at P-height greater than H.

    Up to overall multiplication by a monomial, the coefficients of f along L are

    b0+b1x++bmxm
    for some m and b0,,bm. The mutability condition gives that
    b0+b1x++bmxm=(1+x)rHnei=0scixi, 3.4
    for some coefficients ci. A primitive T-cone with lattice height r contains exactly h lattice points at lattice height h, if 0<h<r, and so if LP contains no residual points and no points outside Ce then m<rHne. In this case (3.3) gives that b0==bm=0, and statement () holds. Otherwise (3.3) holds with s0 and s+1 equal to the total number of points in LP that are either residual or lie outside Ce. Statement () for h=H follows if we can solve (3.3) uniquely for ci in terms of the coefficients bk0,,bks of points on L that are either residual or lie outside Ce. 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 μw,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 RP for P and RQ for Q, noting that, because singularity content is a mutation invariant, |RP|=|RQ|. We have

    f=vPNcvxvandg=vQNc¯vxv,
    where the coefficients cv are affine–linear functions of {cwwRP} and the coefficients c¯v are affine–linear functions of {c¯wwRQ}. The coefficients of the mutation μw,a(f) are affine–linear functions of {cvvPN}, and hence of {cwwRP}. Taking the coefficients of μw,a(f) at elements of RQ gives an affine–linear isomorphism between {cwwRP} and {c¯wwRQ}, and under this isomorphism μw,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, GFGf by construction. Since F is maximally mutable it follows that F is obtained from f by specialization of coefficients.

    This also proved:

    Corollary 3.10.

    Let f be a maximally mutable Laurent polynomial in two variables. Each vertex of the mutation graph Gf is n-valent, where (n,B) is the singularity content of Newt(f).

    Example 3.11.

    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.

    Figure 6.

    Figure 6. A portion of the mutation graph of the rigid MMLP f=y+((1+x)2/xy).

    Figure 7.

    Figure 7. A portion of the quiver mutation graph for Newt(f).

    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 mutation-equivalence 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 πf of a rigid MMLP f matches with the regularized quantum period G^X of the del Pezzo surface ([4], §G). Summarizing:

    Theorem 3.12.

    Mutation-equivalence classes of rigid MMLPs in two variables correspond one-to-one with qG-deformation families of smooth del Pezzo surfaces.

    Combining [30] with [31] proves the analogous result for del Pezzo surfaces with 13(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 YfP1 as follows. Both f and the unit monomial determine sections of the anti-canonical divisor on YP, where P=Newt(f) and YP is the toric variety defined by the normal fan to P. Resolving singularities of YP and resolving basepoints of the rational map YPP1 defined by [1:f] defines the pencil YfP1.

    Theorem 3.13 ([32], theorem 3.13).

    Let f be a maximally mutable Laurent polynomial in two variables. Then a general member of the family of curves YfP1 has genus equal to one plus the number of residual points in P.

    Writing p for the map YfP1, we can consider the monodromy about 0P1 of the local system R1p!Q. Equivalently, this is the monodromy about zero of the Picard–Fuchs differential operator Lf that annihilates the classical period πf.

    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 πf be its classical period. The monodromy about zero of Lf 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 P1×P1 and for the Hirzebruch surface F1. 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:

    πf(t)=1+4t2+36t4+400t6+
    and
    πg(t)=1+2t2+6t3+6t4+60t5+110t6+.

    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).

    4. Some three-dimensional results

    The Minkowski ansatz [1] is extremely successful at recovering mirrors for the 98 three-dimensional 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], table 1).

    (ii)

    The ansatz produces 67 classical periods that are not the quantum period for any three-dimensional Fano manifold ([1], §7).

    (iii)

    The ansatz is not closed under mutation. That is, there exist Laurent polynomials f mutation-equivalent to a Minkowski polynomial that are not themselves Minkowski polynomials. This holds even if one restricts attention to Laurent polynomials with reflexive Newton polytopes.

    Rigid MMLPs have none of these drawbacks. In this section, we report on extensive computational experiments, which in particular give a computer-assisted proof of the following Theorem. A key ingredient here is an effective algorithm for computing the set of MMLPs with given Newton polytope, which we will describe elsewhere [33].

    Theorem 4.1.

    Mutation-equivalence classes of rigid MMLPs f such that Newt(f) is a three-dimensional reflexive polytope correspond one-to-one to the 98 deformation families of three-dimensional Fano manifolds with very ample anti-canonical bundle. Furthermore, each of the 105 deformation families of three-dimensional Fano manifolds has a rigid MMLP mirror.

    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 anti-canonical 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 πf 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 non-reflexive Newton polytopes there exist rigid MMLPs that are not mirror to a Fano manifold.

    Example 4.2.

    Consider the Hilbert series with ID 20522 in the database of possible Hilbert series of Q-factorial terminal Fano threefolds [35,36]. Not all such potential Hilbert series are realized by genuine threefolds, but this one is: it arises from a complete intersection X3,3P(15,2). The variety X3,3 has a terminal singularity of type 12(1,1,1). A Laurent polynomial mirror to X3,3, computed via [11,37], is

    f=z(1+x+y)3(1+1xyz)318.
    This is readily seen to be a rigid MMLP.

    Example 4.3.

    Let PNQ be the canonical Fano polytope with ID 498784 in the database of toric canonical Fano threefolds [38,39]:

    P:=conv{(1,1,1),(1,1,1),(1,3,1),(1,1,0),(1,1,0),(1,1,0),(0,0,1)}.
    This supports a unique rigid MMLP given by
    f=(yxz+1xy)(1+x+y)2+z2.
    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
    (101121011100).
    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(112,22) and has two terminal singularities of type 12(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.

    5. Remarks on higher dimension

    Every known mirror to a Fano manifold in dimension four or more is a rigid MMLP [5,4244]. 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 XP 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 admits3 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|KX| 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.

    Footnotes

    1 In general to specify a GIT quotient V//G, where V is a vector space, we would also need to specify a character χ of G called the stability condition. In our situation, there is a canonical choice for χ, given by B1++Br.

    2 See, for example, ([25], §3.3) or ([26], proposition 4.6).

    3 Comparing birational classifications of singular del Pezzo surfaces [31,49] with classifications of polygons up to mutation [25,50] suggests that, at least in the surface case, a Fano variety X is of class TG if and only if h0(KX)0.

    Acknowledgements

    Much of this paper is based on research guided by and joint work with Alessio Corti. We thank him, Sergey Galkin, Vasily Golyshev, Liana Heuberger, Andrea Petracci and Thomas Prince for many useful conversations. Much of this paper was written during two visits of K.T. to A.K. in 2014, the first funded by a Scheme 4 Grant from the London Mathematical Society and the second funded by EPSRC Institutional Sponsorship EP/N508822/1. Conjecture 5.1 arose from conversations with Corti and Golyshev at the workshop Motivic Structures on Quantum Cohomology and Pencils of CY Motives at the Max Planck Institute for Mathematics, Bonn, in September 2014. The computations underlying this work were performed using the Imperial College High Performance Computing Service and the compute cluster at the Department of Mathematics, Imperial College London; we thank Andy Thomas and Matt Harvey for invaluable technical assistance. We thank John Cannon and the Computational Algebra Group at the University of Sydney for providing licences for the computer algebra system Magma.

    Electronic supplementary material is available online at https://doi.org/10.6084/m9.figshare.c.5647036.

    Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.

    References