The aim of this article is to show that the transpose-dual pairs in the sense
of Ebeling-Ploog of singularities $(Z_{1,0},\, Z_{1,0})$, $(U_{1,0},\,
U_{1,0})$, $(Q_{17},\, Z_{2,0})$, $(W_{1,0},\, W_{1,0})$ that are concluded to
be polytope-dual by the author are actually lattice-dual.Comment: 16 pages,3 tables,to appear in manuscripta mathematic

Mase, Makiko

English

arXiv

Makiko Mase

立教大学学術リポジトリ

COMMENTARII MATHEMATICIUNIVERSITATIS SANCTI PAULIVol. 65, No. 2 2016ed. RIKKYO UNIV/MATHIKEBUKURO TOKYO171–8501 JAPANPolytope Duality for Families of K3 SurfacesAssociated to Transpose DualitybyMakiko MASE(Received July 20, 2016)(Revised October 9, 2016)Abstract. We consider whether or not transpose-dual pairs, which is a Berglund–Hu¨bsch mirror studied by Ebeling and Ploog [3], extend to a polytope duality that has apotential to be lattice dual.1. IntroductionIsolated singularities in C3 are classified by Arnold [1] among which there are classescalled bimodal and unimodal. Our notation follows that of Arnold’s. Not only the classifi-cation, Arnold also finds that there is a duality among unimodal singularities that is calledArnold’s strange duality. The duality is also related to toric geometry and lattice theory.Ebeling and Ploog [3] find an analogous duality concerning bimodal and other singularities,which is actually a Berglund–Hu¨bsch mirror.Batyrev’s proposal [2] of polar duality of reflexive polytopes gives a breakthroughin a construction of mirror partner for toric Calabi-Yau hypersurfaces and later completeintersections.Being origined in physics, there appear a numerical meanings of “mirror” such ascohomological mirror, among which in this article we focus on a relation between Ebel-ing and Ploog’s transpose duality and Batyrev’s polytope duality associating with bimodalsingularities in some manner.In a series of recent studies, it is concluded that transpose-dual pairs (Q12, E18),(Z1,0, E19), (E20, E20), (Q2,0, Z17), (E25, Z19), (Q18, E30) of singularities can extendto a lattice duality by the author [6] following an extension to polytope duality by the authorand Ueda [5]. However, those pairs in the list (∗) below fail to extend to a lattice duality inspite of the fact that they are polytope dual.(∗) (Z13, J3,0), (Z1,0, Z1,0), (Z17, Q2,0), (U1,0, U1,0), (U16, U16),(Q17, Z2,0), (W1,0, W1,0), (W17, S1,0), (W18, W18), (S17, X2,0) .More precisely, for each pair one obtains in [5] reflexive polytopes [MU ] and ′[MU ] sat-isfying that the polar dual of [MU ] is isomorphic to ′[MU ] and that [MU ] and ′[MU ]2010 Mathematics Subject Classification: Primary 14J28; Secondary 14M25.Key words and phrases: K3 surfaces, toric varieties.131132 M. MASErespectively contains the Newton polytope of a compactification polynomial of the definingpolynomial of singularities. Despite this fact it is concluded in [6] that the correspond-ing pairs of families F[MU] and F′[MU] of K3 surfaces are not lattice dual, that is, thePicard lattices Pic([MU ]) and Pic(′[MU ]) of these families do not satisfy an isometryPic([MU ])⊥K3  U ⊕ Pic(′[MU ]). Moreover, for these pairs we can observe that therestriction map H 1,1(P˜[MU], Z) → H 1,1(Z˜, Z) for the minimal model of any genericmember Z ∈ F[MU] is not surjective.The aim of the study is to consider the following problem arisen by Professor Ashik-aga’s question:PROBLEM. Let ((B, f ), (B ′, f ′)) be a transpose-dual pair in the list (∗) togetherwith their defining polynomials f and f ′. Determine whether or not it is possible to takepolynomials F and F ′ that are respectively compactifications of f and f ′, and a reflexivepolytope  such that the following condition (∗∗) holds:(∗∗) F ⊂ , F ′ ⊂ ∗, and L0() = 0 .Here, F and F ′ denote respectively the Newton polytopes of F and of F ′, and ∗ is thepolar dual polytope of .The main theorem of this paper is stated as follows:MAIN THEOREM (Theorem 3.1). For each of the following pairs(Z1,0, Z1,0), (U1,0, U1,0), (Q17, Z2,0), (W1,0, W1,0) ,there exist compactifications F, F ′ and reflexive polytopes  and ′ such that(∗∗) ∗  ′, F ⊂ , F ′ ⊂ ′, and rankL0() = 0hold. Moreover, ρ() + ρ(′) = 20.It can be conjectured that there do not exist reflexive polytopes for pairs (Z13, J3,0),(Z17, Q2,0), (U16, U16), (W17, S1,0), (W18, W18), (S17, X2,0) of singularities satisfyingthe condition (∗∗). We leave the judgement about this conjecture to a further study in thefurure.Section 2 is devoted to recall some facts as to a polytope duality associated to singu-larities. The proof of the main theorem is given in Theorem 3.1 in section 3, where weexplicitely give compactifications and reflexive polytopes for these pairs.Acknowledgment. The author would be grateful to Professor T. Ashikaga for hisquestion of the problem that motivated this article after publication of [6], to Professor N.Aoki who was reading through the first draft carefully and making many helpful sugges-tions, and to Professor M. Kobayashi for his comments and encouragement.Polytope Duality for Families of K3 Surfaces Associated to Transpose Duality 1332. PreliminaryRecall that a Gorenstein K3 surface is a compact complex connected 2-dimensionalalgebraic variety S with at most ADE singularities satisfying KS ∼ 0 and H 1(S, OS) = 0.If a Gorenstein K3 surface is nonsingular, we simply call it a K3 surface.Let M  Z3 be a 3-dimensional lattice and N = HomZ(M, Z)  Z3 the dual of Mwith a natural pairing 〈 , 〉 : N × M → Z. Let  be a 3-dimensional polytope, that is,  isa convex hull of finitely-many points in M ⊗Z R. The associated toric 3-fold is denoted byP. The polar dual ∗ of  is defined by∗ = {y ∈ N ⊗Z R | 〈y, x〉 ≥ −1 for all x ∈ } .Let us recall a toric description of weighted projective spaces. Let a = (a0, a1, a2, a3)be a well-posed quadruple of natural numbers and d = a0 + a1 + a2 + a3. Define a3-dimensional lattice M˜ byM˜ := {(i, j, k, l) ∈ Z4 | a0i + a1j + a2k + a3l ≡ 0 mod d}  Z3 .Note that the lattice M˜ is one-to-one corresponding to the set of monomials of weighteddegree d: indeed, for each (i, j, k, l) ∈ M˜ , a monomial Xi0Xj1Xk2Xl3 is of weighted degreed . Here, the weight of Xi is ai for i = 0, 1, 2, 3. Besides, by letting a be a convex hullof all primitive lattice vectors in M˜ , the associated projective toric 3-fold is the weightedprojective space of weight a.The introduction of reflexive polytope in [2] is motivated by mirror symmetry.DEFINITION 2.1 ([2]). Let  be an integral polytope that contains the origin in itsinterior. The polytope  is called reflexive if its polar dual ∗ is also integral.Not only in a context of mirror, this notion is basically friendly with K3 surfaces asfollows:THEOREM 2.1 ([2]). Let  be a 3-dimensional polytope.(1) The followings are equivalent:(i) The polytope  is reflexive.(ii) The toric 3-fold P is Fano, in particular, general anticanonical members of Pare Gorenstein K3.(2) General anticanonical members of P are simultaneously resolved by a toric (crepant)desingularization of P to be K3 surfaces.Denote for a reflexive polytope  by F a family of (Gorenstein) K3 surfacesparametrised by the complete anticanonical linear system |−KP |. For a member Z inF, denote by Z˜ and P˜ the minimal models in a cause of the simultaneous resolution.In the article, we define that a member Z ∈ F is generic if the following two condi-tions are satisfied:(1) Z is -regular. (See [2] for detail)(2) The Picard group of Z˜ is generated by irreducible components of the restrictionsof the generator of the Picard group of P˜.134 M. MASEIt is proved in [2] that -regularity is a general condition. The condition (2) is alsoa general condition. Note that all Picard lattices of the minimal models of any genericmembers are isometric.DEFINITION 2.2. (1) The Picard lattice Pic() of the family F is the Picard lat-tice of the minimal model of a generic member.(2) ρ() := rank Pic() is called the Picard number of the family F.(3) Let r : H 1,1(P˜, Z) → H 1,1(Z˜, Z) be the restriction mapping of the cohomologygroup. The cokernel of r is denoted by L0().In [5], a notion of transpose duality [3] for singularities is extended to a polytopeduality in the sense of the following theorem :THEOREM 2.2 ([5]). Let ((B, f ), (B ′, f ′)) be a transpose-dual pair together withtheir defining polynomials f and f ′ that are respectively compactified to polynomials Fand F ′. Then, there exist reflexive polytopes [MU ] and ′[MU ] such that∗[MU ]  ′[MU ], F ⊂ [MU ], and F ′ ⊂ ′[MU ] .However, it depends on the pairs that whether or not rankL0([MU ]) = 0 holds. Insection 3, we shall show that some pairs in the list (∗) do have this property.We end this section by giving formulas that are needed in the proof of the main theo-rem. See [4] for details. For a 3-dimensional reflexive polytope , denote by [1] the setof all edges of , and for an edge  ∈ [1], the dual edge in the polar dual polytope ∗is denoted by ∗. The number of lattice points on an edge  is denoted by l(), whilstl() − 2 by l∗(). We haverankL0()=∑∈[1]l∗()l∗(∗) .(1)ρ()=∑∈[1]l(∗) − 3 .(2)Note that rankL0() = rankL0(∗) by the formula.3. Main resultThe chief aim of this section is to prove the following statements.THEOREM 3.1. For pairs (B, B ′) of singularities, if one takes compactificationsF, F ′ as in Table 1, and polytopes , ′ as in Table 2, then,(i)  and ′ are reflexive,(ii) ∗ is isomorphic to ′ up to lattice isometry of Z3,(iii) F ⊂ , and F ′ ⊂ ′ hold, and(iv) rankL0() = 0.Moreover, ρ() + ρ(′) = 20.Proof. Z1,0 case. The defining polynomials of singularities B = Z1,0 and B ′ = Z1,0are the same f = f ′ = x5y + xy3 + z2.Polytope Duality for Families of K3 Surfaces Associated to Transpose Duality 135TABLE 1. Compactifications of singularitiesB F F ′ B ′Z1,0 X5Y + XY 3 + Z2 + W 10X2 X5Y + XY 3 + Z2 + W 14 Z1,0U1,0 X3Y + Y 2Z + Z3 + WX4 XZ3 + X2Y + Y 3 + W 9 U1,0Z2,0 X5Z + XY 3 + Z2 + W 7Y X5Y + WY 3 + XZ2 + W 7 Q17W1,0 X6 + Y 2Z + Z2 + W 6Z X6 + Y 2Z + Z2 + W 12 W1,0TABLE 2. Polytopes that make the pairs polytope dualB vertices of  vertices of ′ B ′Z1,0⎧⎪⎪⎨⎪⎪⎩(−1, 0, 1), (−1, 0, 0),(0, 1,−1), (2, 3,−1),(2, 2,−1), (1,−1,−1),(0,−1,−1)⎫⎪⎪⎬⎪⎪⎭⎧⎪⎪⎨⎪⎪⎩(0, 2,−1), (−1, 1,−1),(−1,−1,−1), (5,−1,−1),(4, 0,−1), (1, 0, 0),(−1,−1, 1)⎫⎪⎪⎬⎪⎪⎭ Z1,0U1,0⎧⎨⎩(−1, 0, 2), (0, 1, 0),(1, 2,−1), (1, 1,−1),(0,−1, 0), (0,−1,−1)⎫⎬⎭⎧⎪⎪⎨⎪⎪⎩(1, 0,−1), (0,−1,−1),(−1,−1,−1), (−1, 2,−1),(1, 2,−1), (1, 0, 1),(0,−1, 2), (−1,−1, 2)⎫⎪⎪⎬⎪⎪⎭ U1,0Z2,0⎧⎨⎩(−1,−1, 2), (0,−1, 0),(1,−1, 0), (1,−1, 1),(1, 2,−3), (0, 0,−1)⎫⎬⎭⎧⎨⎩(−1, 2,−1), (−1,−1, 1),(−1,−1,−1), (6,−1,−1),(2, 1,−1), (0,−1, 1)⎫⎬⎭ Q17W1,0⎧⎨⎩(−1, 0, 1), (−1, 0, 0),(1, 2,−1), (2, 3,−1),(0,−1, 0)⎫⎬⎭⎧⎨⎩(−1,−1,−1), (5,−1,−1),(1, 3,−1), (−1, 3,−1),(−1,−1, 1)⎫⎬⎭ W1,0Take a compactification of f as F = W 10X2 + X5Y + XY 3 + Z2 in the weightedprojective space P(1, 2, 4, 7). Note that F is a different compactification from the onein [3].Take a compactification of f ′ as F ′ = W 14 + X5Y + XY 3 + Z2 in the weightedprojective space P(1, 2, 4, 7). Note that F ′ is the same compactification as in [3].The polytope  contains the Newton polytope of F : indeed, by taking a basis e1 =(−6, 1, 1, 0), e2 = (2, 1,−1, 0), e3 = (−7, 0, 0, 1) for R3, one can see that monomialsW 10X2, X5Y, XY 3, Z2 are respectively corresponding to vertices(0, 1,−1), (2, 2,−1), (1,−1,−1), (−1, 0, 1) .The polytope ′ contains the Newton polytope of F ′: indeed, by taking a standardbasis e′1 = (−2, 1, 0, 0), e′2 = (−4, 0, 1, 0), e′3 = (−7, 0, 0, 1) for R3, one can see that136 M. MASEmonomials W 14, X5Y , XY 3, Z2 are respectively corresponding to vertices(−1,−1,−1), (4, 0,−1), (0, 2,−1), (−1,−1, 1) .The dual polytope ′∗ of ′ is a convex hull of vertices(0, 0, 1), (−1,−2,−3), (−1,−3,−5), (1,−1,−1), (1, 0, 0), (0, 1, 0), (−1,−1,−3)that is mapped to isomorphically from  by a transformation of R3 by the matrixM :=⎛⎝ 1 2 30 −1 −11 2 4⎞⎠that is, (x, y, z)M = (x ′, y ′, z′) for (x, y, z) ∈  and (x ′, y ′, z′) ∈ ′.Therefore,  and ′ are reflexive and the pair is polytope dual.By the formula (1), one gets rankL0() = rankL0(∗) = 0 because for all edgesin  satisfy l∗()l∗(∗) = 0. In fact, at least either  or ∗ has no lattice points in itsinterior.By the formula (2), one can compute thatρ() = 17 − 3 = 14, ρ(∗) = 9 − 3 = 6thus one hasρ() + ρ(∗) = 20 .U1,0 case. The defining polynomials of singularities B = U1,0 and B ′ = U1,0 aref = x3y + y2z + z3, f ′ = x ′z′3 + x ′2y ′ + y ′3, respectively.Take a compactification of f as F = WX4 + X3Y + Y 2Z + Z3 in the weightedprojective space P(1, 2, 3, 3). Note that F is a different compactification from the onein [3].Take a compactification of f ′ as F ′ = W ′9 + X′Z′3 + X′2Y ′ + Y ′3 in the weightedprojective space P(1, 3, 3, 2). Note that F ′ is the same compactification as in [3].The polytope  contains the Newton polytope of F : indeed, by taking a basis e1 =(−5, 1, 1, 0), e2 = (1, 1,−1, 0), e3 = (−3, 0, 0, 1) for R3, one can see that monomialsWX4, X3Y , Y 2Z, Z3 are respectively corresponding to vertices(1, 2,−1), (1, 1,−1), (0,−1, 0), (−1, 0, 2) .The polytope ′ contains the Newton polytope of F ′: indeed, by taking a standardbasis e′1 = (−3, 1, 0, 0), e′2 = (−3, 0, 1, 0), e′3 = (−2, 0, 0, 1) for R3, one can see thatmonomials W ′9, X′Z′3, X′2Y ′, Y ′3 are respectively corresponding to vertices(−1,−1,−1), (0,−1, 2), (1, 0,−1), (−1, 2,−1) .The dual polytope ′∗ of ′ is a convex hull of vertices(0, 0, 1), (−1, 0, 0), (−1, 1, 0), (0, 1, 0), (1, 0, 0), (0,−1,−1)that is mapped to isomorphically from  by a transformation of R3 by the matrixM =⎛⎝ 2 2 1−1 0 01 1 1⎞⎠Polytope Duality for Families of K3 Surfaces Associated to Transpose Duality 137that is, (x, y, z)M = (x ′, y ′, z′) for (x, y, z) ∈  and (x ′, y ′, z′) ∈ ′.Therefore,  and ′ are reflexive and the pair is polytope dual.By the formula (1), one gets rankL0() = rankL0(∗) = 0 because for all edgesin  satisfy l∗()l∗(∗) = 0. In fact, at least either  or ∗ has no lattice points in itsinterior.By the formula (2), one can compute thatρ() = 20 − 3 = 17, ρ(∗) = 6 − 3 = 3thus one hasρ() + ρ(∗) = 20 .Z2,0 and Q17 case. The defining polynomials of singularities B = Z2,0 and B ′ = Q17are f = x5z + xy3 + z2, f ′ = x5y + y3 + xz2, respectively.Take a compactification of f as F = W 7Y + X5Z + XY 3 + Z2 in the weightedprojective space P(1, 1, 3, 5). Note that F is the same compactification as in [3].Take a compactification of f ′ as F ′ = W 7 + X5Y + WY 3 + XZ2 in the weightedprojective space P(1, 1, 2, 3). Note that F ′ is the same compactification as in [3].The polytope  contains the Newton polytope of F : indeed, by taking a basis e1 =(−3, 3, 0, 0), e2 = (−8, 0, 1, 1), e3 = (−6, 1, 0, 1) for R3, one can see that monomialsW 7Y , X5Z, XY 3, Z2 are respectively corresponding to vertices(0, 0,−1), (1,−1, 1), (1, 2,−3), (−1,−1, 2) .The polytope ′ contains the Newton polytope of F ′: indeed, by taking a standardbasis e′1 = (−1, 1, 0, 0), e′2 = (−2, 0, 1, 0), e′3 = (−3, 0, 0, 1) for R3, one can see thatmonomials W 7, X5Y, WY 3, XZ2 are respectively corresponding to vertices(−1,−1,−1), (4, 0,−1), (−1, 2,−1), (0,−1, 1) .The dual polytope ′∗ of ′ is a convex hull of vertices(−1,−3,−4), (0,−2,−3), (0, 1, 0), (1, 0, 0), (0, 0, 1), (−1,−2,−3)that is mapped to isomorphically from  by a transformation of R3 by the matrixM :=⎛⎝ 1 1 11 3 41 2 3⎞⎠that is, M(x, y, z) = (x ′, y ′, z′) for (x, y, z) ∈  and (x ′, y ′, z′) ∈ ′.Therefore,  and ′ are reflexive and the pair is polytope dual.By the formula (1), one gets rankL0() = rankL0(∗) = 0 because for all edgesin  satisfy l∗()l∗(∗) = 0. In fact, at least either  or ∗ has no lattice points in itsinterior.By the formula (2), one can compute thatρ() = 18 − 3 = 15, ρ(∗) = 8 − 3 = 5thus one hasρ() + ρ(∗) = 20 .138 M. MASEW1,0 case. The defining polynomials of singularities B = B ′ = W1,0 are the samef = f ′ = x6 + y2z + z2.Take a compactification of f as F = X6+Y 2Z+Z2+W 6Z in the weighted projectivespace P(1, 2, 3, 6). Note that F is a different compactification from the one in [3].Take a compactification of f ′ as F ′ = X′6 + Y ′2Z′ + Z′2 + W ′12 in the weightedprojective space P(1, 2, 3, 6). Note that F ′ is the same compactification as in [3].The polytope  contains the Newton polytope of F : indeed, by taking a basis e1 =(−5, 1, 1, 0), e2 = (1, 1,−1, 0), e3 = (−6, 0, 0, 1) for R3, one can see that monomialsX6, Y 2Z, Z2, W 6Z are respectively corresponding to vertices(2, 3,−1), (0,−1, 0), (−1, 0, 1), (−1, 0, 0) .The polytope ′ contains the Newton polytope of F ′: indeed, by taking a standardbasis e′1 = (−2, 1, 0, 0), e′2 = (−3, 0, 1, 0), e′3 = (−6, 0, 0, 1) for R3, one can see thatmonomials X′6, Y ′2Z′, Z′2, W ′12 are respectively corresponding to vertices(5,−1,−1), (−1, 1, 0), (−1,−1, 1), (−1,−1,−1) .The dual polytope ′∗ of ′ is a convex hull of vertices(0, 1, 0), (−1,−1,−3), (0,−1,−2), (1, 0, 0), (0, 0, 1)that is mapped to isomorphically from  by a transformation of R3 by the matrixM :=⎛⎝ 1 1 30 0 −11 2 3⎞⎠that is, M(x, y, z) = (x ′, y ′, z′) for (x, y, z) ∈  and (x ′, y ′, z′) ∈ ′.Therefore,  and ′ are reflexive polytopes and the pair is polytope dual.By the formula (1), one gets rankL0() = rankL0(∗) = 0 because for all edgesin  satisfy l∗()l∗(∗) = 0. In fact, at least either  or ∗ has no lattice points in itsinterior.By the formula (2), one can compute thatρ() = 21 − 3 = 18, ρ(∗) = 5 − 3 = 2thus one hasρ() + ρ(∗) = 20 .References[ 1 ] Arnol’d, V. I. Critical points of smooth functions and their normal forms. Russian Math. Surveys 30, No. 5(1975), 1–75.[ 2 ] Batyrev, V. Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties. J. Alg.Geom. 3 (1994), 493–535.[ 3 ] Ebeling, W., and Ploog, D. A geometric construction of Coxeter-Dynkin diagrams of bimodal singulari-ties. Manuscripta Math. 140 (2013), 195–212.[ 4 ] Kobayashi, M. Duality of weights, mirror symmetry and Arnold’s strange duality. Tokyo J. Math. 31(2008), 225–251.Polytope Duality for Families of K3 Surfaces Associated to Transpose Duality 139[ 5 ] Mase, M. and Ueda, K. A note on bimodal singularities and mirror symmetry. Manuscripta Math. (online31 August 2014) 146, 1–2 (2015), 153–177.[ 6 ] Mase, M. A mirror duality for families of K3 surfaces associated to bimodular singularities. ManuscriptaMath. (online 26 September 2015) 149, 3–4 (2016), 389–404.Makiko MASEDepartment of Mathematics, Rikkyo University3–34–1 NishiIkebukuro, Toshimaku,Tokyo 171–8501, Japane-mail: mtmase@arion.ocn.ne.jp

Lattice duality for families of $K3$ surfaces associated to transpose duality

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Lattice duality for families of K3K3K3 surfaces associated to transpose duality

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Lattice duality for families of $K3$ surfaces associated to transpose duality