On the Twelve-Point Theorem for $\ell$-Reflexive Polygons

It is known that, adding the number of lattice points lying on the boundary of a reflexive polygon and the number of lattice points lying on the boundary of its polar, always yields 12. Generalising appropriately the notion of reflexivity, one shows that this remains true for"$\ell$-reflexive polygons". In particular, there exist (for this reason) infinitely many (lattice inequivalent) lattice polygons with the same property. The first proof of this fact is due to Kasprzyk and Nill. The present paper contains a second proof (which uses tools only from toric geometry) as well as the description of complementary properties of these polygons and of the invariants of the corresponding toric log del Pezzo surfaces.


Introduction
The purpose of this paper is to give a second proof of the so-called "Twelve-Point Theorem" for "reflexive polygons" (see below Theorem 1.27), to explain where 12 comes from by taking a slightly different approach, and to provide some additional consequences of it from the point of view of toric geometry.
• Polygons. Let P ⊂ R 2 be a (convex) polygon, i.e., the convex hull conv(A) of a finite set A ⊂ R 2 of at least 3 non-collinear points. We denote by Vert(P ) and Edg(P ) the set of its vertices and the set of its edges, respectively, and by ∂P and int(P ) its boundary and its interior, respectively. If the origin 0 ∈ R 2 belongs to int(P ), then its polar polygon is defined to be P • := x ∈ R 2 x, y ≥ −1, ∀y ∈ P , where x, y := x 1 y 1 + x 2 y 2 , for x = x1 x2 ∈ R 2 and y = y1 y2 ∈ R 2 , is nothing but the usual inner product. Since 0 ∈ int(P • ) and (P • ) • = P, the polarity induces bijections and with v, v denoting the vertices of F.
• Lattices. Since we shall deal with a special sort of lattice polygons, we first recall some basic properties of lattices (cf. (i) N is a discrete subgroup of the additive group R 2 (i.e., n − n ∈ N for all n, n ∈ N, and for every n ∈ N there is an ε ∈ R >0 , s.t. B ε (n) ∩ N = {n}, where B ε (n) := x ∈ R 2 x − n ≤ ε ), and N spans the entire R 2 as R-vector space.
If P 1 N P 2 , we say that P 1 and P 2 are equivalent up to N -umimodular transformation. If P ∈ POL 0 (N ), we denote by [P ] N := { R ∈ POL 0 (N )| R N P } its equivalence class. Definition 1.9. If P ∈ POL 0 (N ), then for a fixed basis matrix B ∈ GL 2 (R) of N we have N = Φ B (Z 2 ) with Φ B ∈ Aut(R 2 ). Thus, we may define the polygon . If the induced bijection POL 0 (N )/ N [P ] N −→ [P st ] Z 2 ∈ POL 0 (Z 2 )/ Z 2 is taken into account, it is sometimes convenient to work with the equivalence class of P st instead of that of P (and with the standard lattice Z 2 instead of N ), e.g., when we draw figures, when we construct certain polygon classification lists etc. It is worth mentioning that Ehr N (P ; k) = Ehr Z 2 (P st ; k) for all k ∈ Z ≥0 , because (∂P ∩ N ) = (∂P st ∩ Z 2 ) and area N (P ) = area Z 2 (P st ).
• LDP-polygons. Let N ⊂ R 2 be a lattice. A point n ∈ N {0} is said to be primitive (ii) Let Q ⊂ R 2 be an LDP-polygon (w.r.t. N ) and M := Hom Z (N, Z) be the dual of our reference lattice N. For F ∈ Edg(Q) we denote by η F ∈ M the unique primitive lattice point which defines an inward-pointing normal of F. The affine hull of F is of the form y ∈ R 2 −η F , y = l F , for some positive integer l F . This l F is nothing but the integral distance between 0 and F, the so-called local index of F (w.r.t. Q). The index of Q is defined to be the positive integer := lcm{ l F | F ∈ Edg(Q)} . It is easy to prove that = min {k ∈ Z >0 |Vert(kQ • ) ⊂ M } . This can be derived by using (more general) results of Hensley [34], and Lagarias & Ziegler [50]. LDPpolygons are of particular interest because their N -classes parametrise the isomorphism classes of toric log del Pezzo surfaces. (See below §5.) LDP-triangles of index ≤ 3 have been classified (up to unimodular transformation) in [17, §6] and [18]. More recently, this classification has been extended considerably in [45] via a certain algorithm, by means of which it is possible to produce the LDP-polygons of given index (up to unimodular transformation) by fixing a "special" edge and performing a prescribed successive addition of vertices. Of course, their cardinality grows rapidly as we increase indices! The classification is complete for ≤ 17. Useful details for the structure of each of these 16 + 30 + · · · + 1892 = 15346 LDP-polygons (vertices of representatives of standard models w.r.t. a suitable coordinate system, interior and boundary lattice points, area, local indices, Ehrhart and Hilbert series etc.) are included in the database [8].
Proof. By the definition of and by (1.6), the equivalences (i)⇔(ii)⇔(iii) are obvious. For (v)⇔(i) see Hibi [35, §35], [36]. The implication (v)⇒(iv) is apparently true (by applying (v) for k = 0). It suffices to show the validity of implication (iv)⇒(ii). Assuming that int(Q) ∩ N = {0}, (1.5) gives Let F be an edge of Q having n, n as vertices. We observe that l F = contradicting (1.7). Thus, l F = 1 for all F ∈ Edg(Q).  (ii) The notion of reflexivity is extendable to lattice polytopes of any dimension ≥ 3 via conditions 1.13 (i), (iii) and (v) (which remain equivalent). It was introduced by Batyrev in [3, §4]. Condition 1.13 (iv) is necessary (for a lattice polytope of dimension ≥ 3 to be reflexive) but is not sufficient: There are several lattice polytopes of dimension ≥ 3 which have the origin as the only interior lattice point without being reflexive. (Reflexive polytopes play a pivotal role in the so-called "combinatorial mirror symmetry"; cf. [15,Chapters 3 and 4] and [5]). On the other hand, in dimension 2 we meet nice lattice point enumerator identities like (1.8). (1.8) One proof consists in case-by-case verification of (1.8) by passing through the explicit classification of reflexive polygons, i.e., by the so-called "exhaustion method".
Theorem 1.17 (Classification of reflexive polygons). Let (Q, N ) be a reflexive pair. Then Q has at most 6 vertices, and a representative of the equivalence class of its standard model (w.r.t. any basis matrix of N ) is exactly one of the sixteen Z 2 -polygons Q 1 , . . . , Q 16 illustrated in Figure 1, whose vertices (in an anticlockwise order) are given in the second columns of the tables: (1.9) Figure 1.
Proof of Theorem 1.16 via Theorem 1.17. One checks directly that (1.10) and that the vertices of the polars of Q 7 , . . . , Q 10 are the following: The entries of the third columns of tables (1.9), combined with (1.10) and (1.11), give (iii) In fact, as it follows from the work of Batyrev mentioned in (i), the most natural interpretation for the presence of the number 12 in (1.8) arises from the application of the celebrated Noether's formula for the Euler-Poincaré characteristic of the structure sheaf of the minimally desingularized compact toric surface which is associated with a reflexive polygon. (See also [16,Theorem 10.5.10, pp. 510-511] and Note 7.5 below.) Max Noether [55] discovered this remarkable formula in 1870's in the framework of the theory of adjoints of algebraic surfaces. For comments on its early history and for a modern direct proof see Gray [29, §2], Hulek [42, §3] and Piene [58]. Besides, Noether's formula can be viewed, alternatively, as the Hirzebruch-Riemann-Roch formula [40, p. 154] in (complex) dimension 2. (The coefficient of t 2 in the expansion of t exp(t)−1 as a Taylor series equals 1/12.) (iv) Poonen & Rodriguez-Villegas [59] added two new proofs of Theorem 1.16: (a) by stepping into the space of reflexive polygons, and (b) by exploiting basic properties of the universal cover of SL 2 (R) and of the corresponding modular (cusp) forms of weight 12. (See also Castryck [10, §2].) Elementary geometric proofs (using reduction to the parallelogram case and Scott's inequality [66], respectively) are due to Cencelj . This particular choice of representatives from each of the 16 available equivalence classes is such that 0 1 ∈ Vert(Q i ) for all i ∈ {1, ..., 16}, and will be convenient for the formulation of Theorem 6.9.
As will be seen in the sequel, there are no -reflexive polygons having more than 6 vertices and there are no -reflexive polygons with even (see Corollaries 7.6 and 7.8). For the time being, taking into account the precise polygon data from [8] we deduce the following: 23. The values of the enumerating functions → (RP ν ( ; N )) and → (RP( ; N )) for ν ∈ {3, 4, 5, 6} and for odd ≤ 25 are those given in the table: Proof. Since the affine hull of every F ∈ Edg(Q) is of the form y ∈ R 2 η F , y = − , we have i.e., Vert(Q * ) consists of primitive lattice points, and 0 ∈ int(Q • ) ⊆ int(Q * ). Since the affine hull of any edge of Q * is of the form x ∈ R 2 x, −n = for some n ∈ Vert(Q), the integral distance between 0 and the edge equals . Hence, Q * is an LDP-polygon of index w.r.t. M. Note 1.26. In analogy to (1.1), (1.2), we establish bijections: (1.13) and where n, n denote the vertices of F.
(ii) If 3 , then is an -reflexive quadrilateral with and is an -reflexive pentagon with and (iv) The hexagon is -reflexive having For = 3 this is illustrated in Figure 2. (Here, the = 3 case gives an interesting example, because the associated toric log del Pezzo surface is the only log del Pezzo surface among those with Fano index 1, anticanonical degree ≥ 2 and singularities of type (2, 3) (in our notation), the regular locus of which has non-trivial fundamental group.  Since does not admit upper bound, the "exhaustion method" is apparently not the right method to verify formula (1.15). The first proof of Theorem 1.27 given by Kasprzyk and Nill in [46] is purely combinatorial, clear and short, and makes use of the so-called -reflexive loops. Nevertheless, it offers no essential information about the connection with the "classical"approach mentioned in 1.18 (iii). In [46, §1.6] the question was raised, whether there is also another direct argument arising from algebraic geometry in the case of -reflexive polygons. Here, maintaining the technique of lattice change from [46, §2.2] in our toolbox, we shall provide such an argument and a second proof of Theorem 1.27: Its disadvantage lies in that it is by no means short (as one has to translate everything into the language of toric varieties, and this requires several steps). On the other hand, among its main advantages are included: (a) Noether's formula remains again at least one assured reason for the appearance of 12 (in combination with other useful formulae in the > 1 case), and (b) several other results are obtained by transferring the duality concept from the -reflexive polygons to the corresponding log del Pezzo surfaces. More precisely, the paper is organised as follows: In Section 2 we focus on the two non-negative, relatively prime integers p = p σ and q = q σ parametrising the N -cones σ and characterising the twodimensional toric singularities. Moreover, we describe briefly the minimal desingularization procedure by means of the negative-regular continued fraction expansion of q q−p and by determining the exceptional prime divisors after the Hilbert basis computation of the corresponding cone. In section 3 we recall the interrelation between lattice polygons and compact toric surfaces with a fixed ample divisor, and explain how one computes the area and the number of lattice points lying on the boundary of a lattice polygon via intersection numbers. (See Theorems 3.8 and 3.9.) In §4- §5 we indicate the manner in which we classify (up to isomorphism) compact toric surfaces via the wve 2 c-graphs and, in particular, toric log del Pezzo surfaces via LDP-polygons. Giving priority to those log del Pezzo surfaces which are associated with -reflexive polygons we present in §6 the geometric meaning of the lattice change from [46, §2.2] (which, in a sense, seems to be the standard method of reducing -reflexivity to 1-reflexivity): One may patch together canonical cyclic covers over the singularities in order to construct a finite holomorphic map of degree and to represent the surfaces under consideration as global quotients of Gorenstein del Pezzo surfaces by finite cyclic groups of order . Results of this geometric interpretation (e.g., Proposition 6.13, concerning the relation between the self-intersection numbers of the canonical divisors), combined with Noether's formula and other information derived from the desingularization, give rise to a new proof of Theorem 1.27 in §7 and to various consequences of it (upper bound for (Vert(Q)), a proof of "oddness" of , a new approach of Suyama's formula, number-theoretic identities involving types of singularities, combinatorial triples, Dedekind sums etc.). In section 8 we discuss certain new phenomena which occur in the > 1 case, and give typical examples. For instance, the characteristic differences no longer vanish (as in the = 1 case, where each 1 -reflexive polygon has only the origin in its interior), but are equal to the number of lattice points lying on the boundary of I(Q * ) and I(Q), i.e., of the polygons defined as convex hulls of the (at least 4, non-collinear) interior lattice points of Q * and Q, respectively. Finally, in §9 we verify (in the lowest dimension) the existence of a large number of families of combinatorial mirror pairs (of certain smooth curves of high genus, owing to this new wider framework of duality) and in §10 we state some open questions. We use tools from discrete and classical toric geometry (adopting the standard terminology from [16], [23], [26], and [56]), and some basic facts and formulae from intersection theory (see [ 2. Two-dimensional lattice cones and toric surfaces • N -cones. A 2-dimensional strongly convex polyhedral cone in R 2 (with the origin 0 ∈ R 2 as its apex and x 1 , x 2 ∈ R 2 {0} as generators) is a subset σ of R 2 of the form where x 1 , x 2 are R-linearly independent, and σ ∩ (−σ) = {0}.
(ii) If ε is the sign of det(n 1 , n 2 ) and M σ : , then {n 1 , n 1 } is a basis of N, and n 2 = pn 1 + qn 1 with q = mult N (σ). The above integers p =: p σ and q =: q σ associated with σ do not depend on the particular choice of B.
On the other hand, since Therefore, q = q σ does not depend on the choice of B. In addition, if B is another basis matrix of N, then B = BA for some A = ( a11 a12 a21 a22 ) ∈ GL 2 (Z) (see Proposition 1.4). Let ≡ p (mod q) , and p = p σ is also independent of the choice of B.
Note 2.5. It should be stressed that p = p σ does depend on which minimal generators of σ is regarded as first and which as second (because of the defining conditions (2.2) and (2.3)). For this reason, by writting σ = R ≥0 n 1 + R ≥0 n 2 , with n 1 , n 2 as its minimal generators, their ordering will always be implicit (and p σ well-defined). Proposition 2.7 gives the precise description of what happens by interchanging n 1 and n 2 or, more generally, by replacing σ with a τ N σ.
Proof. Let σ st , τ st be the standard models of σ, τ w.r.t. an arbitrary basis matrix B of the lattice N, and M σ , M τ ∈ GL 2 (Z) the corresponding matrices defined in 2.4 (iii), so that In the first case det(M τ AM −1 σ ) has to be equal to 1, which means that q τ = q σ and p τ − p σ ≡ 0(mod q), i.e., p τ = p σ (because 0 ≤ p σ , p τ < q σ = q τ ). In the second case, det(M τ AM −1 σ ) = −1, i.e., q τ = q σ and 1 − p σ p τ ≡ 0(mod q) ⇒ p τ = p σ . Definition 2.8. Let σ be an N -cone. Since the two integers p = p σ and q = q σ associated with σ (by Proposition 2.4) parametrise uniquely the equivalence class [σ] N up to replacement of p by its socius p, we shall henceforth say that σ is of type (p, q) (or simply that σ is a (p, q)-cone).
• Hibert basis. σ ∩ N is an additive commutative monoid for any N -cone σ. It is known (by Gordan's Lemma [56, Proposition 1.1 (iii), p. 3]) that σ ∩ N is finitely generated (as a semigroup), and that among all generating systems there is a system Hilb N (σ) of minimal cardinality, the so-called Hilbert basis of σ, which is uniquely determined (up to reordering of its elements) by the following characterisation: n cannot be expressed as sum of two other vectors belonging to σ ∩ (N {0}) .
• Affine toric surfaces. Let σ be an N -cone and M the dual lattice. We set Ce(m) be the C-algebra with basis {e(m) |m ∈ S σ } consisting of formal elements which fulfill the exponential law: Since S σ is finitely generated (as a semigroup), Definition 2.11. We denote by U σ (or, more precisely, by U σ,N , whenever it is necessary to stress that σ is an N -cone) the affine toric surface Spec(C[S σ ]) which is associated with σ and has C[S σ ] as coordinate ring. (U σ is a 2-dimensional normal complex analytic variety embedded in C k as vanishing locus of finitely many binomials which generate I ; see [56, Proposition 1.2, pp. [4][5]. To work with the embedding of U σ into an affine complex space of minimal dimension it is enough to replace the arbitrary Next, we use the identifications with (a) induced by

and (c) by
The standard action of the algebraic torus on (the set of points Hom semigr. (S σ , C) of) U σ can be conceived as multiplication of semigroup homomorphisms: We denote by orb(σ) ∈ Hom semigr. (S σ , C) (or, more precisely, by orb N (σ), whenever it is necessary to stress that σ is an N -cone) the unique point of U σ remaining fixed under (2.6), i.e., the semigroup homomorphism mapping m ∈ S σ onto 1 whenever m, y = 0 for all y ∈ σ, and onto 0 otherwise. By Propositions 2.7 and 2.12 the type (p, q) of σ (up to replacement of p by its socius p) determines the isomorphism class of the germ (U σ , orb (σ)) .
(ii) The minimal generators of σ constitute a basis of N.
Proof. Since q = mult N (σ) (by 2.4 (iii)), q = 1 if and only if the triangle having the origin and the two minimal generators of σ as vertices does not contain any additional lattice point (see (2.1)). Hence, the equivalence of (i) and ( If the conditions of Proposition 2.13 are satisfied, then σ is said to be a basic N -cone. The non-basic N -cones are characterised by the following: Proposition 2.14. Let σ be an N -cone of type (p, q). If q > 1, then p ≥ 1 and orb(σ) ∈ U σ is a cyclic quotient singularity. (It is often called cyclic quotient singularity of type 1 (p, q).) In particular, is the cyclic group of order q which is generated by diag(ζ −p q , ζ q ) (with ζ q := exp(2π √ −1/q)) and acts on (i) orb(σ) ∈ U σ is a Gorenstein singularity (i.e., O Uσ,orb(σ) is a Gorenstein local ring).
• N -fans. A set ∆ consisting of finitely many N -cones and their 0-and 1-dimensional faces (i.e., the origin and their rays) will be referred to as an N -fan if for any N -cones σ 1 , σ 2 belonging to ∆ with σ 1 = σ 2 , the intersection σ 1 ∩ σ 2 is either the singleton {0} or a common ray of σ 1 and σ 2 . (We shall denote by ∆ (1) and ∆ (2) the set of rays and the set of N -cones of ∆, respectively, and by |∆| the support of ∆, i.e., the union of its elements.) If ∆ is an N -fan, then using the so-called Glueing Lemma (for the affine toric surfaces U σ , σ ∈ ∆ (2)) one defines the (normal) toric surface X(N, ∆) associated with ∆. The actions of the algebraic torus (2.5) on the affine toric surfaces U σ , σ ∈ ∆ (2) , defined in (2.6) are compatible with the patching isomorphisms, giving the natural action of T on X(N, ∆) (which extends the multiplication in T). All the orbits w.r.t. it are either of the form orb(σ), σ ∈ ∆ (2) , with orb(σ) the unique T-fixed point of U σ → X(N, ∆) as defined before, or of the form orb( ) := Hom gr. ( ⊥ ∩ M, C {0}), ∈ ∆ (1) (which are 1-dimensional subvarieties of X(N, ∆)) with M := Hom Z (N, Z) and ⊥ := x ∈ R 2 x, y = 0, ∀y ∈ , or, finally, orb({0}) = T. If D is a Weil divisor on X(N, ∆), then D ∼ D for some T-invariant Weil divisor D (with "∼" meaning linearly equivalent). It is known that every Weil divisor on X(N, ∆) is a Q-Cartier divisor (see [26, p. 65]). We denote by Div T W (X(N, ∆)) and Div T C (X (N, ∆)) the groups of T-invariant Weil and Cartier divisors on X(N, ∆), respectively. The first of them is described as follows: with n denoting the minimal generator of ∈ ∆ (1) .
The group Div T C (X (N, ∆)) can be also described in terms of ∆-support functions. A ∆-support function is a group isomorphism having   [26, p. 59]).
Let ∆ be an N -fan. If ∆ is a refinement of ∆ (i.e., if ∆ is an N -fan with |∆ | = |∆| and each N -cone of ∆ is a union of N -cones of ∆ ), then the induced T-equivariant holomorphic map is proper and birational (see [56,Corollary 1.17,p. 23]). The singular locus of X(N, ∆) is In the case in which Sing(X(N, ∆)) = ∅, it is always possible to construct (by suitable successive N -cone subdivisions) a refinement ∆ of ∆ such that Sing(X(N, ∆ )) = ∅, i.e., such that (2.13) is a resolution of the singular points of X(N, ∆) (a desingularization of X(N, ∆)). The so-called minimal desingularization f : X(N, ∆) −→ X(N, ∆) of a toric surface X(N, ∆) (which is unique, up to factorisation by an isomorphism) is that one arising from the coarsest refinement ∆ of ∆ with Sing(X(N, ∆)) = ∅.
• Intersection numbers. If X(N, ∆) is smooth, then the intersection number D 1 · D 2 ∈ Z of two divisors D 1 , D 2 on X(N, ∆) with compactly supported intersection is defined in the usual sense (see [27, 2.4.9, p. 40]). If X(N, ∆) is singular and compact, and D 1 , D 2 two Q-Weil divisors on X(N, ∆), we set • Continued fractions and minimal desingularization of U σ . Let σ = R ≥0 n 1 + R ≥0 n 2 be an Ncone with n 1 , n 2 as minimal generators. The affine toric surface U σ can be viewed as If σ is non-basic of type (p, q) (as in Proposition 2.14), we consider the negative-regular continued fraction expansion of q q−p (with b s ≥ 2) and define recursively u 0 , u 1 , . . . , u s , u s+1 ∈ N by setting It is easy to calculate b 1 , b 2 , . . . , b s (see, e.g., [19, Lemma 3.4 (i)] and [43]) and to verify that Note 2.18. (i) p, p, q and the sum b 1 + · · · + b s are related to each other via the formula , considering the negative-regular continued fraction expansion It is known (cf. [56, p. 29]) that

22)
and denote by ∂Θ cp σ (resp., by ∂Θ cp σ ∨ ) the part of the boundary ∂Θ σ (resp., of ∂Θ σ ∨ ) containing only its compact edges, then of the N -fan ∆ σ (having the Hilbert basis elements of σ as minimal generators of its rays) contains only basic N -cones, and constitutes the coarsest refinement of ∆ σ with this property. Therefore, it gives rise to the construction of the minimal T-equivariant resolution Proof. See Hirzebruch [39, pp. 15-20] who constructs X(N, ∆ σ ) by resolving the unique singularity lying over the origin of C 3 in the normalisation of the hypersurface (z 1 , = 0 , and Oda [56, pp. 24-30] for a proof which uses only tools from toric geometry.

Compact toric surfaces and lattice polygons
Let N be a lattice in R 2 and M := Hom Z (N, Z). An N -fan ∆ is said to be complete whenever |∆| = R 2 . • Nef and ample Cartier divisors on compact toric surfaces. From now on we shall work with a fixed complete N -fan ∆. For a given D ∈ Div T C (X (N, ∆)), we set with h D ∈ SF(N, ∆) as defined in (2.11). (We write P D,∆ instead of P D if we wish to emphasise which our reference fan is.) P D is bounded and its affine hull has dimension ≤ 2. Moreover, there is a unique We denote by for all y, y ∈ R 2 and t ∈ [0, 1] and SUCSF (N, ∆) := h ∈ UCSF (N, ∆) for all σ 1 , σ 2 ∈ ∆ (2) with σ 1 = σ 2 , h| σ1 , h| σ2 are different linear functions the sets of upper convex and strictly upper convex ∆-support functions, respectively.
, then the following conditions are equivalent: is generated by its global sections.
(viii) D is "nef" (numerically effective), i.e., the intersection number of D with any (irreducible compact) curve on X (N, ∆) is non-negative.
, then the following conditions are equivalent: Proof. The equivalence of the conditions (i), (ii), (iii) Proof. This follows directly from the highest power term in the (generalised) Riemann-Roch formula: For details see [ • Lattice polygons and normal fans. For given D ∈ Div T C (X (N, ∆)) we have defined P D = P D,∆ in (3.1) which is an M -polygon whenever D is ample. Starting with an M -polygon P one can, conversely, construct a compact toric surface X (N, Σ P ) and a distinguished ample divisor D P on it.
Definition 3.6. Let P be an M -polygon. For every m ∈ Vert(P ) we define the M -cone It is easy to see that is a complete N -fan. Σ P is called the normal fan of P. Denoting by η F ∈ N the (primitive) inwardpointing normal of an F ∈ Edg(P ) (cf. 1.10 (ii)) we observe that where F, F are the edges of P having m as their common vertex. Now writing P in the form we set Proposition 3.7. D P is ample and its support function h P := h D P : R 2 → R (often called the support function of P ) is defined as follows: Next, we consider the set of pairs Theorem 3.9. Let P be an M -polygon. If f : X(N, Σ P ) −→ X(N, Σ P ) is the minimal desingularization of X(N, Σ P ), then the pullback f (D P ) of the ample divisor D P is the unique nef divisor on X(N, Σ P ) for which P = P D P = P f (D P ) (or, more precisely, P = P D P ,Σ P = P f (D P ), Σ P ), and for which Proof. For the first assertion see [16, Proposition 6.2.7, p. 281]. (D P has strictly upper convex support function and therefore f (D P ) has upper convex support function, and P = P D P ,Σ P = P f (D P ), Σ P because by Theorem 3.2 the M -polygon associated with a nef divisor is determined by its support function.) Now let k be an arbitrary non-negative integer.
Thus, the Euler-Poincaré characteristic is computed via (3.9) and (3.5) as follows: and (3.6) is therefore true. On the other hand, Riemann-Roch Theorem for the projective smooth toric surface X(N, Σ P ) gives  .2) for the divisor f (D P ).)

wve 2 c-graphs and classification up to isomorphism
Given two complete N -fans ∆, ∆ , under which conditions are the corresponding compact toric surfaces X (N, ∆) and X (N, ∆ ) biholomorphically equivalent, i.e., isomorphic in the analytic category? The answer to this question requires the use of the so-called "wve 2 c-graphs", the weights of which are the types of the N -cones of ∆ and ∆ , and some additional characteristic integers determined by the minimal desingularizations of X (N, ∆) and X (N, ∆ ) (see below Theorem 4.5). Let ∆ be a complete N -fan, and let σ i = R ≥0 n i + R ≥0 n i+1 , i ∈ {1, . . . , ν}, be its N -cones (with ν ≥ 3 and n i primitive for all i ∈ {1, . . . , ν}), enumerated in such a way that n 1 , . . . , n ν go anticlockwise around the origin exactly once in this order. (Convention. We set n ν+1 := n 1 and n 0 := n ν . In general, in definitions and formulae involving enumerated sets of numbers or vectors in which the index set {1, . . . , ν} is meant as a cycle, we shall read the indices i "mod ν", even if it is not mentioned explicitly.) By (2.7) we have Suppose that σ i is a (p i , q i )-cone for all i ∈ {1, . . . , ν} and introduce the notation to separate the indices corresponding to non-basic from those corresponding to basic N -cones. Obviously,  and, finally, define the complete N -fan ∀i ∈ {1, . . . , ν}, the exceptional divisor replacing the singular point orb(σ i ) via f, and where each of the K(E (i) )'s is a Q-Cartier divisor (the local canonical divisor of X(N, ∆) at orb(σ i ) in the sense of [17, §1]) supported in the union N,∆) ), then f is said to be crepant. where and with I ∆ , J ∆ as in (4.2). The triples (p i , q i , r i ), i ∈ {1, . . . , ν}, will be referred to as the combinatorial triples of ∆.  A circular graph is a plane graph whose vertices are points on a circle and whose edges are the corresponding arcs (on this circle, each of which connects two consecutive vertices). We say that a circular graph G is Z-weighted at its vertices and double Z-weighted at its edges (and call it wve 2 c-graph, for short) if it is accompanied by two maps assigning to each vertex an integer and to each edge a pair of integers, respectively. To the complete N -fan ∆ (as described above) we associate an anticlockwise directed wve 2 c-graph G ∆ with (v ν+1 := v 1 ), by defining its "weights" as follows: The reverse graph G rev ∆ of G ∆ is the directed wve 2 c-graph which is obtained by changing the double weight (p i , q i ) of the edge v i v i+1 into ( p i , q i ) and reversing the initial anticlockwise direction of G ∆ into clockwise direction (see Figure 4).
∼ = " indicates graph-theoretic isomorphism (i.e., a bijection between the sets of vertices which preserves the corresponding weights). For further details and for the proof of Theorem 4.5 see [17, §5]. (Conventions for the drawings. When we draw concrete wve 2 c-graphs in the plane we attach, for simplification's sake, only the weight −r i at v i without mentioning v i itself, for i ∈ {1, . . . , ν}, and the double weight (p i , q i ) at the edge v i v i+1 , for i ∈ I ∆ , while we leave edges v i v i+1 , i ∈ J ∆ , without any decoration in order to switch to the notation for the Z-weighted circular graphs introduced by Oda in [56, pp. 42-46] which are used for the study of smooth compact toric surfaces.)

Toric log del Pezzo surfaces
A compact complex surface is called log del Pezzo surface if (a) it has at worst log-terminal singularities, i.e., quotient singularities, and (b) there is a positive integer multiple of its anticanonical divisor which is a Cartier ample divisor. The index of a log del Pezzo surface is defined to be the least positive integer having property (b). Every smooth compact toric surface possesses a unique anticanonical model (in the sense of Sakai [63]) which has to be a toric log del Pezzo surface; and conversely, every toric log del Pezzo surface is the anticanonical model of its minimal desingularization (see [17, Theorem 6.5, p. 106]). Proof. Suppose that := min k ∈ Z >0 −kK X(N,∆) ∈ Div T C (X(N, ∆)) and is ample . By Theorem 2.16 and (2.12), − K X(N,∆) = ( means that there is a unique set { l σ | σ ∈ ∆ (2)} ⊂ M := Hom Z (N, Z) such that l σ , n = − for ∈ ∆ (1) ∩ σ. From the implication (i)⇒(iv) in Theorem 3.3 (applied for the divisor D = − K X(N,∆ Q ) ) we deduce that P − K X(N,∆) is an M -polygon with vertex set Vert(P − K X(N,∆) ) = { l σ | σ ∈ ∆ (2)} (without repetitions). We observe that the polygon 1 P − K X(N,∆) := conv 1 l σ | σ ∈ ∆ (2) contains 0 in its interior. Since 1 l σ , n = −1 for ∈ ∆ (1) ∩ σ, its polar polygon is where Q st , Q st are the standard models of Q, and Q , respectively, w.r.t. B. It is a easy to verify that this is equivalent to Thus, (ii)⇔(i) can be seen to be true by making use of Theorem 4.5.
Q (as defined in 1.10 (ii)) is given by the formula .   The wve 2 c-graph G ∆ Q * of its dual (1.17) is shown in Figure 6, where for ≥ 7 we set :  For the wve 2 c-graph G ∆ Q * of its dual (1.19) see Figure 8.  The wve 2 c-graph G ∆ Q * of its dual (1.21) is given in Figure 10. (iv) The wve 2 c-graph G ∆ Q of the -reflexive hexagon (1.22) is shown in Figure 11. Note that for its dual (1.23) we have G ∆ Q * ∼ = gr.
6. Lattice change and cyclic covering trick whenever > 1 • Degree. Let f : X −→ Y be a proper holomorphic map between two complex (analytic) spaces. f is called finite if it is closed (as map) and for every y ∈ Y the fibre f −1 ({y}) consists of finitely many points. • Analytic spectrum. Let X be a complex space and G be an arbitrary sheaf of O X -modules (an O X -module, for short). G is said to be of finite type at x ∈ X if there is an open neighborhood U x of x and a G| Ux -epimorhism O κx X → G| Ux for a positive integer κ x . G is called of finite type on X if it is of finite type at all points x ∈ X. G is coherent if G is of finite type on X and, in addition, for every x ∈ X and every finite system s 1 , ..., s κ ∈ G(U x ) of sections over an open neighborhood U x of x the sheaf of relations Rel x (s 1 , ..., s κ ) (which is the kernel of the G| Ux -homomorhism O κ Ux → G| Ux determined by s 1 , ..., s κ ) is of finite type at x. If G happens to be a sheaf of O X -algebras (an O X -algebra, for short), i.e., if G x is an O X,x -module and at the same time endowed with a ring structure for all x ∈ X, then the following is of particular importance. Theorem 6.1. Let X be a complex space and G be a coherent O X -algebra. Then there exists a unique (up to analytic isomorphism) complex space Specan(G), the so-called analytic spectrum of G, as well as a finite holomorphic map π : Specan(G) −→ X, such that (i) there is an isomorphism π * (O Specan(G) ) ∼ = G, and (ii) there is a bijection π −1 (x) ↔ Max-Spec(G x ) between the set of points of the fibre of π over x and the set of maximal ideals of the stalk of G at x, for all x ∈ X.
For a rough local description of this "spectrum" in the analytic category we refer to [25, pp. 59-62] and [48, 45.B.1, p. 172], and for more details on the construction and the main properties of π to Houzel [41].
• Normal complex varieties which are Q-Gorenstein. If X is a normal complex variety, then its Weil divisors can be described by means of "divisorial" sheaves. (ii) If X 0 is a non-singular open subvariety of X with codim X X X 0 ≥ 2, then F | X 0 is invertible and where ι : X 0 → X denotes the inclusion map.
The divisorial sheaves are exactly those satisfying the above conditions. Since a divisorial sheaf is torsion free, there is a non-zero section s ∈ H 0 (X, M X ⊗ O X F), with H 0 (X, M X ⊗ O X F) ∼ = C (X) · s, and F can be considered as a subsheaf of the constant sheaf M X of meromorphic functions of X, i.e., as a special fractional ideal sheaf. Let M * X and O * X be the sheaves of germs of not identically vanishing meromorphic functions and of nowhere vanishing holomorphic functions on X, respectively.
is a bijection, and induces a Z-module isomorphism. In fact, to avoid torsion, one defines this Z-module structure by setting for any Weil divisors D, D 1 , D 2 and j ∈ Z.
Let now Ω 1 Reg(X) be the cotangent sheaf on Reg(X) := X Sing(X) ι → X, and for j ≥ 2 let us set Ω j Reg(X) := j Ω 1 Reg(X) . The canonical divisor K X of X is that one, the class of which is mapped by d onto the canonical divisorial sheaf ω X := ι * Ω dim C (X) Reg(X) . Note that ω X = ω [1] X := O X (K X ) and that ω [j] Reg(X) ) ⊗j ) for all j ∈ Z.
Definition 6.4. X is called Q-Gorenstein if its canonical divisorial sheaf ω X = O X (K X ) is such that K X is Q-Cartier divisor. If X is Q-Gorenstein, then we set index(X):= min{j ∈ Z ≥1 |jK X is Cartier }.
• Canonical cyclic coverings. Given a point x 0 of a normal Q-Gorenstein complex variety X, we consider an affine neighborhood U of x 0 representing the set germ at x 0 , and a nowhere vanishing section is the natural map, then the coherent O U -module equipped with the multiplication " " being induced by setting The pair (U can , π U ) has (and is up to an analytic isomorphism uniquely determined by) the following properties: (i) U can is a normal complex variety and the fiber π −1 U ({x 0 }) over x 0 is a singleton (say {y 0 }). (ii) The field extension C(U can ) of C(U ) is Galois with Galois group G U ∼ = Z/(index(U ))Z and with a generator g of G U acting on R U as follows: i.e., K U can is a Cartier divisor, and U can is a Q-Gorenstein affine complex variety of index 1.
(v) There is a non-vanishing section s ∈ H 0 (U can , O U can (K U can )) around the point y 0 which is semiinvariant w.r.t. the action of G U and on which G U acts faithfully.
(π U : U can −→ U is said to be the canonical cyclic cover of U of degree deg(π U ) = index(U ).) Remark 6.6. (i) In particular, π U : U can −→ U is surjective 3 and can be viewed as the quotient map by an appropriate identification U ∼ = U can /G U .
of C(U ) has G U as Galois group, and • Back to our specific toric log del Pezzo surfaces. Let be an integer > 1, and (Q, N ), (Q * , M ) two -reflexive pairs, where M := Hom Z (N, Z), with X(N, ∆ Q ) and X(M, ∆ Q * ) the corresponding toric log del Pezzo surfaces. Assume that n 1 , . . . , n ν are the vertices of Q ordered anticlockwise, and for i ∈ {1, . . . , ν} define F i := conv({n i , n i+1 }) to be the i-th edge of Q (as in 5.4 (ii)) and σ i := σ Fi = R ≥0 n i + R ≥0 n i+1 the N -cone of type (p i , q i ) supporting F i . It is easy to verify that and can be viewed as the quotient map by the identification Since Now we apply Theorem 6.5 for U σi,N . For every j ∈ {0, 1, ..., l Fi −1} the divisor −jK U σ i ,N is T N -invariant and Γ(U σi,N , ω N ) is an affine toric surface which is Q-Gorenstein and of index 1 (which means that it is a two-dimensional Gorenstein variety 4 ), and the canonical cover map (6.1) is equivariant. Setting ϕ i : of C(U σi,N ) has a cyclic Galois group, say is a Gorenstein toric affine surface, it suffices for our purposes to recall that it has to appear as the quotient of C 2 by a finite cyclic subgroup H i of SL 2 (C) acting diagonally. W.l.o.g. we may assume that Specan(R U σ i ,N ) ∼ = C 2 /H i ∼ = U σi,Li (i.e., the toric affine surface associated with the same cone σ i but with respect to another lattice L i ⊂ R 2 , such that |L i : Using the equivariant holomorphic map determined by the dotted arrow in the diagram: On the other hand, the restriction is anétale holomorphic map (and, in particular, a topological, i.e., an unramified covering map), and (where π 1 (...) denotes the fundamental group of these pathwise connected spaces, cf. [52, Theorem 2.8, p. 18]). Furthermore, the composite of theétale holomorphic maps is the universal cover of U σi,Li {orb Li (σ i )} which is simply connected) gives the following short exact sequence of fundamental groups: = |G i | , and we conclude that H i = G i and L i = Λ Fi . Finally, it is by construction obvious that the orbit orb Λ F i (σ i ) ∈ U σi,Λ F i ∼ = C 2 /G i is either a smooth point (whenever G i is trivial) or a cyclic quotient singularity of type (1, qi Now let Λ Q ⊆ N be the sublattice generated by the boundary lattice points of Q and Λ Q * ⊆ M be the sublattice generated by the boundary lattice points of Q * . In addition, (Q, Λ Q ) and (Q * , The "beauty" of being -reflexive is mainly embodied in the following property: All local indices of the edges F i of Q coincide with the index of the toric log del Pezzo surface X(N, ∆ Q ), and this allows us to patch together the canonical cyclic covers over the affine neighborhoods of its singularities in order to create a single global finite holomorphic map π Q of degree and represent X(N, ∆ Q ) as a global quotient space. Theorem 6.9. There is an equivariant (w.r.t. the actions of the algebraic tori T Λ Q and T N ) finite holomorphic map which has degree and coincides with the quotient map by the identification , and Q ♦ := Φ B ♦ −1 (Q). Hence, the dotted arrow (which denotes the T Z 2 -equivariant holomorphic map induced by Φ A ,k ) in the following diagram can be viewed again as a quotient map.
Proof. Since Q is -reflexive, we have l Fi = and U σi,Λ F i = U σi,Λ Q , and for the canonical cyclic covers π U σ i ,N which are constructed by Lemma 6.7 we obtain for all i ∈ {1, ..., ν}. Since U σi,Λ Q i ∈ {1, ..., ν} is an open covering of X(Λ Q , ∆ Q ), we may patch them together by setting π Q is by definition a finite holomorphic map of degree = |N : Λ Q | , with is 7-reflexive, and via A 7,1 we get Φ A7,1 (Q 7 ) = Q. The toric del Pezzo surface X(Z 2 , ∆ Q ) has three cyclic quotient singularities: One of type (5,14), one of type (16,21), and one of type (5,7). X(Z 2 , ∆ Q 7 ) inherits a Gorenstein cyclic quotient singularity of type (1, 2) over the first, a Gorenstein cyclic quotient singularity of type (1, 3) over the second, and a smooth point over the third.
Remark 6.11. Clearly, Theorem 6.9 gives (RP( ; N )) ≤ 16 φ( ), where φ is Euler's totient function, but this is only a rough upper bound. In fact, (RP( ; N )) depends essentially on number-theoretic restrictions on the weights of the possible wve 2 c-graphs. In practice, for the classification of -reflexive polygons and for the construction of precise tables like those in [8], one has to perform ad-hoc tests to distinguish lattice-inequivalent polygons. (Cf. Grinis & Kasprzyk [30] for a more general discussion on the normal forms of lattice polytopes.) Lemma 6.12. Let Y and Z be two normal projective surfaces and π : Y −→ Z be a generically finite and surjective holomorphic map of degree d. If D 1 , D 2 are two Q-Weil divisors on Z, then where π (D j ) is the pullback of D j , j ∈ {1, 2}, via π (in the sense of [27, p. 32]).
Proof. Denoting by ρ : Z −→ Z the minimal desingularization of Z, by δ : Y −→ Z × Z Y the normalisation of the fiber product Z × Z Y, and by γ : Y −→ Y the minimal desingularization of Y , we obtain a commutative diagram of the form: On the other hand, and therefore (6.3) is true.
Proposition 7.4. The self-intersection number of the canonical divisor of X(N, ∆ Q ) is Correspondingly, the self-intersection number of the canonical divisor of X(M, ∆ Q * ) is Proof. Applying Proposition 1.21 for the -reflexive pair (Q * , M ) and formula (3.7) (for P = Q * ) we get which gives (7.1). The proof of (7.2) is similar.
• Consequences of Theorem 1.27. Let be an integer ≥ 1 and let (Q, N ) be an -reflexive pair.
Maintaining the notation introduced above, formula (1.15) gives significant information about Q, Q * , (∂Q ∩ N ) , (∂Q * ∩ M ), , and the combinatorial triples of the corresponding fans ∆ Q , ∆ Q * . First proof. Since the number of the vertices of Q (resp., of Q * ) does not change by passing from lattice N to lattice Λ Q (resp., from M to Λ Q * ), the claim is correct by Theorem 1.17.
(ii) If (∂Q ∩ N ) = 8, then (∂Q * ∩ M ) = 4, which is again impossible (by using the same argument as in case (i) but this time with Q * in the place of Q).
Since the socius p 1 of p 1 is odd too, the last divisibility condition is impossible (because p 1 + p 2 + 1 is an odd integer). By (i), (ii) and (iii) we conclude that is always odd.
• Further interrelations of the data of both sides. The duality established by the bijections (1.13) and (1.14) implies certain additional number-theoretic identities which involve the combinatorial triples of both sides.
Proposition 7.14. If > 1, then for each i ∈ {1, . . . , ν} we have and (7.26) Proof. Since σ i = R ≥0 n i + R ≥0 n i+1 is a (p i , q i )-cone, there exist a basis matrix B of N and a matrix M σi ∈ GL 2 (Z) such that where σ st i is the standard model of σ i w.r.t. B (see Proposition 2.4 and Figure 12). Φ Mσ i B −1 maps n i onto 1 0 and n i−1 onto a point n1,i−1 n2,i−1 ∈ Z 2 , i.e., σ i−1 onto the Z 2 -cone R ≥0 We observe that the point of ∂Θ cp (7.28) Figure 12.
Finally, it remains to give the explicit number-theoretic description of the link between p * i , p i * and the multiplicity q * i , provided that p i and q i are assumed to be known, and, respectively, of the link between p i , p i and the multiplicity q i , provided that p * i and q * i are assumed to be known.
and set and λi := with ε = 1 for ρ even and ε = −1 for ρ odd. Then κ i , λ i ∈ Z and Denoting by z i the unique positive integer which is smaller than q * i and satisfies we obtain if ρ is even and d − 1, if ρ is even and d and [19,Remark 3.2,p. 217]). Assume that (defined by interchanging the ordering of the minimal generators of (7.34)) is a (z i , q * i )-cone. By Proposition 2.4 z i has to be the unique positive integer which is smaller than q * i and satisfies (7.32). Using (7.31) and (7.25) we can write the left-hand side of (7.32) as follows: Thus, (7.33) is true and (7.34) is a ( z i , q * i )-cone (cf. Note 2.5 and the proof of Proposition 2.7). Since both Proof. Since Σ Q * = ∆ Q and − K X(N,∆ Q ) = D Q * , Theorem 3.9 (applied for the lattice M -polygon Q * ) implies that the pullback f (− K X(N, and We define the function h : for the divisor f (D Q * )) and ∆ Q contains only basic N -cones. Thus, by (2.9) and (2.10) (and by the implication (ii)⇒(viii) in Theorem 3.2 for h ), h determines a unique nef divisor D h ∈ Div T C (X(N, ∆ Q )), namely (according to (2.12) for the N -fan ∆ Q ). Since I(Q * ) = x ∈ R 2 x, n ≥ h (n ), ∀ ∈ ∆ Q (1) , (8.1) is true.

Families of combinatorial mirror pairs in the lowest dimension
Batyrev's combinatorial mirror symmetry construction [3] is completely efficient whenever the "ambient spaces" are toric Fano varieties with at worst Gorenstein singularities of (complex) dimension ≥ 4 or at least of dimension 3. In the latter case, the general members of the linear system defined by their anticanonical divisors are K3-surfaces. In the lowest dimension 2 (i.e., when the "ambient spaces" are Gorenstein log del Pezzo surfaces), the corresponding general members are elliptic curves. The generalisation (in dimension 2) which takes place by passing from Gorenstein log del Pezzo surfaces (defined by 1-reflexive polygons) to log del Pezzo surfaces defined by -reflexive polygons leaves little room for the determination of "combinatorial mirrors", and as yet only up to homeomorphism: The corresponding general members are smooth projective curves with Hodge diamond having (as unique non-trivial number) their genus (also called sectional genus) at the left and at the right corner. This genus is > 1 whenever > 1. give curves Proj(S Q * ) ∩ H which are linearly equivalent to − K X(N,∆ Q ) . For generic H's the intersection C Q := Proj(S Q * ) ∩ H is (by Bertini's Theorem) a smooth connected projective curve in the non-singular locus of Proj(S Q * ) ∼ = X(N, ∆ Q ). The genus of g(C Q ) of C Q is called the sectional genus of X(N, ∆ Q ) and will be denoted simply as g Q . and has dimension h 0 (C Q , ω C Q ) := dim C (H 0 (C Q , ω C Q )) = dim C (H 1 (C Q , O C Q )) = g Q (by adjunction). Moreover, C 2 Q = (− K X(N,∆ Q ) ) 2 = 2 K 2 X(N,∆ Q ) =  [11], [12]. In our case, we can assume that the curves C Q are nothing but Zariski closures Z f of affine hypersurfaces Z f ⊂ T N for Laurent polynomials f having Q * as their Newton polygon. It would be interesting, for reflexive -polygons Q, to investigate if (beyond the topological equivalence) there is a deeper relation between (e.g., certain complex structures on) Z f ⊂ X(N, ∆ Q ) and Z g ⊂ X(M, ∆ Q * ) on the "other side". For given combinatorial mirror partners (as defined in 9.4) what would be the connection between their "strict" mirrors (which turn out to be particular 3-dimensional Landau-Ginzburg models) from the point of view of the homological mirror symmetry for curves of high genus? (Cf. Efimov [21].) i.e., a formula which is no longer symmetric w.r.t. to Q and Q • . In particular, if Q happens to be a smooth 8 (also known as Delzant) polytope and d ≥ 3, we have Conjecture B. Suppose that d ≥ 4, Λ F d−2 (Q) (resp., Λ F d−2 (Q * ) ) is the sublattice of N (resp., of M ) generated by the set F d−2 (Q) of the faces of Q (resp., by the set F d−2 (Q * ) of the faces of Q * ) of codimension 2, and that (Q, Λ F d−2 (Q) ) is an 1-reflexive pair. Then (Q * , Λ F d−2 (Q * ) ) is an 1-reflexive pair too, formula (10.2) is true for d = 4 (if one replaces in it F • by F * and Q • by Q * ), and formula (10.3) is true for d ≥ 5 (for both Q and Q * ).
(d) Since the "cyclic covering trick" of Theorem 6.5 is independent of the dimension (and is a standard tool for reducing log terminal and log canonical singularities of a Q-Gorenstein variety, to canonical and, respectively, log canonical singularities of index 1, cf. [51,), in order to tackle the above conjectures, one should come up with analogues of Lemma 6.7, Theorem 6.9, and Proposition 6.13, being valid in dimension d ≥ 3. If d ≥ 3, the singularities of X(N, ∆ Q ) are not necessarily isolated, and one has to construct carefully a suitable stratification of the singular locus. In addition, even the nature of singularities may differ (as it is known that in dimensions ≥ 3 there exist toric singularities which are not quotient singularities). Nevertheless, toric singularities are "relatively mild" singularities and it seems to be not very difficult to deal with them. On the other hand, the analogues of (6.4) in high dimensions should relate various (usual, orbifold or stringy) Chern classes of X(N, ∆ Q ) and X(Λ F d−2 (Q) , ∆ Q ). (Furthermore, it would be desirable if one could keep all the required arguments independent of particular desingularizations of X(N, ∆ Q ).) (iv) Recently, log del Pezzo surfaces have also attracted increasing interest in the framework of the socalled homological mirror symmetry for Fano varieties in dimension d = 2. (See, e.g., [1], [14] and [47], and the references therein.) It was proposed that log del Pezzo surfaces with cyclic quotient singularities admit Q-Gorenstein toric degenerations corresponding (under mirror symmetry) to maximally mutable Laurent polynomials in two variables, and that the quantum period of such a surface coincides with the classical period of its mirror partner. Thus, the combinatorics of mutation and toric deformations (which are closely related to geometric properties of LDP-polygons 11 ) play an important role in the conception of this new approach. It comes into question whether the toric log del Pezzo surfaces associated with -reflexive polygons (perhaps with prescribed singularities) are of particular value for these investigations.