K3 surfaces from configurations of six lines in P 2 and mirror symmetry I
Shinobu Hosono, Bong H. Lian, Hiromichi Takagi, Shing-Tung Yau
aa r X i v : . [ m a t h . AG ] M a r K3 SURFACES FROM CONFIGURATIONS OF SIX LINES IN P AND MIRROR SYMMETRY I
SHINOBU HOSONO, BONG H. LIAN, HIROMICHI TAKAGI AND SHING-TUNG YAU
Abstract.
From the viewpoint of mirror symmetry, we revisit the hypergeo-metric system E (3 , for a family of K3 surfaces. We construct a good resolu-tion of the Baily-Borel-Satake compactification of its parameter space, whichadmits special boundary points (LCSLs) given by normal crossing divisors. Wefind local isomorphisms between the E (3 , systems and the associated GKZsystems defined locally on the parameter space and cover the entire param-eter space. Parallel structures are conjectured in general for hypergeometricsystem E ( n, m ) on Grassmannians. Local solutions and mirror symmetry willbe described in a companion paper [18], where we introduce a K3 analogue ofthe elliptic lambda function in terms of genus two theta functions. Contents Introduction
The hypergeometric system E (3 , P branched along six lines 32.2. Period integrals of X P (3 , M SecP of P (3 , GKZ hypergeometric system from E (3 , E (3 , M , from period integrals M SecP and M , More on the resolutions of M , X ′ → X along the singular locus 185.2. Flipping the line ℓ in e X to P Blowing-up the Baily-Borel-Satake compactification M φ : M , M M S action on M M by open sets of toric varieties 257. Hypergeometric D -modules on Grassmannians Six singular vertices of
Sec ( A ) Four dimensional cones C and C NE Picard-Fuchs operators on
Spec C [( σ (1)2 ) ∨ ∩ L ] Birational map φ : M , M Singular lines in M Properties used in the proof of Theorem 6.12 . Introduction
Consider double covers of P branched along four points in general positions.They define a family of elliptic curves called the Legendre family over the modulispace of the configurations of four points on P , which are naturally parametrizedby the cross ratio of the four points. It is a classical fact that the elliptic lambdafunction is defined as a modular function that arises from the hypergeometric seriesrepresenting period integrals for the Legendre family.Higher dimensional analogues of the Legendre family have been studied in manycontext in the history of modular forms and analysis related to them. Among oth-ers, Matsumoto, Sasaki and Yoshida [22] have studied extensively in the 90’s thetwo dimensional generalization of the Legendre family, i.e., the double covers ofthe projective plane P branched along six lines in general positions. After mak-ing suitable resolutions, the double covers define a family of smooth K3 surfacesparametrized by the configurations of six lines. In [22], the authors studied in greatdetails the period integrals of the family and determined the monodromy proper-ties of the period integrals completely. They described the set of the differentialequations satisfied by the period integrals in terms of the so-called Aomoto-Gel’fandsystem [1, 9, 8] on Grassmannians G (3 , , and named them hypergeometric system E (3 , .Around the same time in the 90’s, period integrals for families of Calabi-Yau man-ifolds were studied intensively to verify several predictions from mirror symmetryof Calabi-Yau manifolds. For Calabi-Yau manifolds given as complete intersectionsin a toric variety, it is now known that the period integrals for such a family aresolutions to a hypergeometric system called Gel’fand-Kapranov-Zelevinski (GKZ)system. In particular, it was shown in [17, 16] that for GKZ systems in this contextthere exist special boundary points called large complex structure limits (LCSLs),and mirror symmetry appears nicely in the form of generalized Frobenius methodwhich provides a closed formula for period integrals and mirror map near theseboundary points.In this paper, we will revisit the hypergeometric system E (3 , from the view-point of mirror symmetry of K3 surfaces. Despite the fact that many analyticproperties of E (3 , have been studied in details in the literature e.g. [22, 25], itwas not clear how to construct the degeneration points (LCSLs) in the parameterspace of E (3 , . We will find that the D -module associated to the hypergeomet-ric system E (3 , over its parameter space is locally trivialized by the D -moduleof the corresponding GKZ hypergeometric system ( Theorem 7.1 ). Thanks tothis general property, it turns out that the techniques developed in [17, 16] forGKZ systems can be applied to E (3 , ( Theorem 7.2 ); this includes the existenceof the degeneration points and the closed formula of the period integrals aroundthem. To show our results, we first cover the parameter space of E (3 , , which canbe identified with the Baily-Borel-Satake compactification of the family of the K3surfaces, by certain Zariski open subsets of toric varieties on which GKZ systemsare defined. Using this covering property, we finally show that there are two nicealgebraic resolutions of the Baily-Borel-Satake compactification ( Theorem 6.12 )which are related by a four dimensional flip.Around the special degeneration points (LCSLs), following [17, 16], we can definethe so-called mirror maps. In our case, these mirror maps can be regarded as two imensional generalizations of the elliptic lambda function. We will call them λ K -functions . In a companion paper [18], we will describe the λ K -functions in termsof genus two theta functions. Moreover, we will find that, corresponding to the twodifferent algebraic resolutions related by a flip, there exist two different definitionsfor the λ K -functions.Here is the outline of this paper: In Section 2, after introducing our family ofK3 surfaces and the hypergeometric system E (3 , satisfied by period integrals,we will introduce the configuration space of six ordered points as the parameterspace of E (3 , . We summarize known properties about the compactification ofthe parameter space of E (3 , and also introduce other closely related parameterspaces: the configuration space of 3 points and 3 lines in P and the parameterspace of the GKZ system which trivializes the E (3 , . In Section 3, we describe atoric compactification of the parameter space of this GKZ system, and constructthe expected LCSLs after making a resolution. In Section 4, we observe that theconfiguration space of 3 points and 3 lines in P arises naturally from certain residuecalculations of a period integral. We find that the toric compactification for theGKZ system gives a toric partial resolution of the GIT compactification of the con-figuration space of 3 points and 3 lines in P . In Section 5 and 6, we reconstructthe partial resolution using classical projective geometry. Transforming this partialresolution (locally) by certain birational map to the Baily-Borel-Satake compact-ification, we construct the desired algebraic resolutions of the Baily-Borel-Satakecompactification. In Section 7, we combine the results of the preceding sectionsand rephrase them in the language of D -modules to state the main results of thispaper. We also formulate conjectural generalizations of our results. Acknowledgements:
This project was initiated by a question by Naichung C.Leung about hypergeometric systems on Grassmannians to S.H. We are grateful tohim for asking the question which actually has drawn our attention to the problemsleft unsolved in the 90’s. We are also grateful to Osamu Fujino for useful advice tothe proof of Claim F.1. S.H. would like to thank for the warm hospitality at theCMSA at Harvard University where progress was made. This work is supported inpart by Grant-in Aid Scientific Research (C 16K05105, S 17H06127, A 18H03668S.H. and C 16K05090 H.T.). B.H.L and S.-T. Yau are supported by the SimonsCollaboration Grant on Homological Mirror Symmetry and Applications 2015–2019.2.
The hypergeometric system E (3 , Double covering of P branched along six lines. Let us consider six lines ℓ i ( i = 1 , .., in P in general position. We denote them by { ℓ i ( x , y , z ) = 0 } ⊂ P with the following linear forms: ℓ i ( x , y , z ) := a i z + a i x + a i y ( i = 1 , ..., . When the lines are in general position, the double cover branched along the sixlines defines a singular K3 surface with A singularity at each 15 intersection points P ij := ℓ i ∩ ℓ j . Blowing-up the 15 A singularities, we have a smooth K3 surface X of Picard number 16 generated by the hyperplane class H from P and the − curves of the exceptional divisors E ij from the blow-up. The configurations of six ines define a four dimensional family of K3 surfaces, which we will call double coverfamily of K3 surfaces for short in this paper. The period integrals of the family ofholomorphic two forms and their monodromy properties were studied extensivelyin [22] by analyzing the hypergeometric system E (3 , . We will revisit the system E (3 , from the viewpoint of mirror symmetry and provide a new perspective formirror symmetry.2.2. Period integrals of X . Recall that the Legendre family consists of ellipticcurves given by double covers of P branched along four points in general position.The double cover family of K3 surfaces is a natural generalization of the Legendrefamily. Analogous to the period integrals of the Legendre family [29, Chap.IV,10]are the period integrals of a holomorphic two form:(2.1) ¯ ω C ( a ) = ˆ C dµ qQ i =1 ℓ i ( x , y , z ) , where dµ = x d y ∧ d z − y d x ∧ d z + z d x ∧ d y and C is an integral (transcendental)cycle in H ( X, Z ). Explicit descriptions of the transcendental cycles can be foundin [22]. Also the lattice of transcendental cycles is determined to be T X ≃ U (2) ⊕ U (2) ⊕ A ⊕ A , where U (2) represents the hyperbolic lattice U of rank 2 with the Gram matrixmultiplied by 2, and A = h− i is the root lattice of sl (2 , C ) . As obvious inthe above definition, the period integrals ¯ ω C ( a ) determine (multi-valued) functionsdefined on the set of × matrices A representing (ordered) six lines in generalpositions. Explicitly, we describe the matrices A by(2.2) A = a a a a a a a a a a a a a a a a a a . Let M , be the affine space of all × matrices, and set M o , := { A ∈ M , | D ( i , i , i ) = 0 (1 ≤ i < i < i ≤ } with D ( i , i , i ) representing × minors of A . Then, under the genericity as-sumption, the configurations of six lines are parametrized by P (3 ,
6) := GL (3 , C ) (cid:31) M o , (cid:30) ( C ∗ ) , where ( C ∗ ) represents the diagonal C ∗ -actions. The differential operators whichannihilate the period integrals define the hypergeometric system of type E (3 , [22,Sect.1.4], which is the Aomoto-Gel’fand system on Grassmannian G (3 , [1, 9, 8].The following proposition is easy to derive. roposition 2.1. The period integral ¯ ω ( a ) satisfies the following set of differentialequations: (2.3) (i) X i =0 a ij ∂∂a ij ¯ ω ( a ) = −
12 ¯ ω ( a ) , ≤ j ≤ , (ii) X j =1 a ij ∂∂a kj ¯ ω ( a ) = − δ ik ¯ ω ( a ) 1 ≤ i ≤ , (iii) ∂ ∂a ij ∂a kl ¯ ω ( a ) = ∂ ∂a il ∂a kj ¯ ω ( a ) , ≤ i, k ≤ , ≤ j, l ≤ . Proof.
The relations (i) and (iii) are rather easy to verify by differentiating (2.1)directly. To derive (ii), we note that dµ = i E d x ∧ d y ∧ d z holds with the Eulervector field E = x ∂∂ x + y ∂∂ y + z ∂∂ z . Since the Euler vector field is invariant underthe linear coordinate transformation, it is easy to verify ¯ ω ( ga ) = (det g ) − ¯ ω ( a ) , for the left GL (3 , C ) -action on A = ( a ij ) ∈ M o , . The relation follows from theinfinitesimal form of this relation. (cid:3) In the paper [22], the hypergeometric functions representing the period inte-grals has been studied in details using the following affine coordinate system of thequotient P (3 , : x x x x . However, this affine coordinate turns out to be inadequate for studying mirrorsymmetry. In particular, in order to construct the special boundary points, calledlarge complex structure limits (LCSLs), we need a suitable compactification.2.3.
Period domain and compactifications of the parameter space P (3 , . Mirror symmetry for two or three dimensional Calabi-Yau hypersurfaces or com-plete intersections in toric Fano varieties was worked out in many examples in the90’s by constructing families of Calabi-Yau manifolds and by studying period inte-grals associated to holomorphic n -forms for n = 2 or . It is now known that thegeometry of mirror symmetry appears, in a certain simplified form [27], near thespecial boundary points which are given as normal crossing boundary divisors insuitable compactifications of the parameter spaces for the families of hypersurfaces[17]. The double cover family of K3 surfaces does not belong to these well-studiedfamilies of Calabi-Yau manifolds. However, its parameter space P (3 , admitsmany nice compactifications relevant to describe the boundary points. We summa-rize several compactifications and describe their relationships. (2.3.a) Period domain D K = Ω( U (2) ⊕ ⊕ A ⊕ ) . Since the generic member X of the double cover family of K3 surfaces has the transcendental lattice T X ≃ U (2) ⊕ ⊕ A ⊕ , the period integral defines a map from P (3 , to the period domain D K := (cid:8) [ ω ] ∈ P (( U (2) ⊕ ⊕ A ⊕ ) ⊗ C ) | ω.ω = 0 , ω. ¯ ω > (cid:9) + , here + represents one of the connected components. Let us denote by G the Grammatrix of the lattice U (2) ⊕ ⊕ A ⊕ given in the following block-diagonal from: G = (cid:18) (cid:19) ⊕ (cid:18) (cid:19) ⊕ ( − ⊕ ( − . Using this, we define G := (cid:8) g ∈ P GL (6 , Z ) | t gGg = G, H ( g ) > (cid:9) with H ( g ) = ( g + g )( g + g ) − ( g + g )( g + g ) , which is a discretesubgroup of Aut( D K ) (see [21, Sect.1.4]). In [22, Prop.2.7.3], it is shown thatthe monodromy group of period integrals coincides with the congruence subgroup G (2) = { g ∈ G | g ≡ E mod } , hence P (3 , ≃ D K / G (2) holds and D K givesthe unifomization of the multi-valued period integral on the configuration space P (3 , . (2.3.b) GIT compactification M . A natural compactification of P (3 , is givenby parameterizing the six lines { ℓ i } by the corresponding points { a i } in the dualprojective space ˇ P and arrange the corresponding ordered six points as in (2.2)with a i = t ( a i , a i , a i ) . The configuration space of these ordered six points is awell-studied object in geometric invariant theory. In [6, 24], one can find that acompactification M is given as a double cover of P branched along the so-calledIgusa quartic, which has the following description:(2.4) M ≃ (cid:8) Y = F ( Y , ..., Y ) (cid:9) ⊂ P (1 , , where F is the quartic polynomial F = ( Y Y s + Y Y − Y Y ) + 4 Y Y Y Y s with Y s := Y − Y + Y + Y − Y . See Appendix D.1 for a brief summary. Since M is a geometric compactification, the (multi-valued) period map from P (3 , to D K naturally extends to M , which we will write P : M → D K . (2.3.c) Baily-Borel-Satake compactifications. In [21], it was shown explicitlythat the double cover M coincides with the Baily-Borel-Stake compactification ofcertain arithmetic quotient of the symmetric space of type I , defined by H = (cid:8) W ∈ M at (2 , C ) | ( W † − W ) / i > (cid:9) , where W † := t W . The Siegel upper half space of genus two h is contained in H as the locus satisfying W = t W . To introduce the arithmetic quotients of H ,following the notation of [21], we define discrete subgroups of Aut( H ) : Γ := (cid:8) g ∈ P GL (4 , Z [ i ]) | g † Jg = J (cid:9) , g † := t ¯ g, J := (cid:0) E − E (cid:1) , Γ T := Γ ⋊ h T i , T : W t W ( W ∈ H ) , Γ M := { gT a ∈ Γ T | ( − a det( g ) = 1 , a = 0 , } . We also introduce the congruence subgroups:
Γ(1 + i ) := { g | g ≡ E mod (1+i) } , Γ T (1 + i ) := Γ(1 + i ) ⋊ h T i . Then the arithmetic group relevant to the quotient is Γ M (1 + i ) := Γ M ∩ Γ T (1 + i ) , hich defines the quotient Γ M (1 + i ) \ H . Note that there is a natural map Γ M (1 + i ) \ H → Γ T (1 + i ) \ H which is generically .The Baily-Borel-Satake compactification of the arithmetic quotient of Γ T (1 + i ) \ H is given explicitly by the Zariski closure of the image of the map Φ : H → P , W [Θ ( W ) , · · · , Θ ( W ) ] , where theta functions Θ i ( W )( i = 1 , ..., correspond to ten different (even) spinstructures. These squares of the theta functions are modular forms of weight twoon the group Γ T (1 + i ) with a character given by determinant det( gT a ) = det( g ) for gT a ∈ Γ T (1+ i ) , ( a = 0 , (see [21, Prop. 3.1.1]). Also, there are five linear relationsamong them. Hence we have Γ T (1 + i ) \ H ≃ P for the compactification. When W = t W , these theta functions reduces to the theta functions θ ( τ ) , ..., θ ( τ ) of genus two which generate Siegel modular forms of level two and even weights.The Igusa quartic is a quartic relation satisfied by θ i ( τ ) , hence defines a quartichypersurface in P .Actually the above five linear relations correspond to Plücker relations (D.2)under a suitable identification of the Θ i ( W ) ’s with the semi-invariants Y k ’s, whichwe will do in our companion paper [18] to introduce λ K -functions. Under thisidentification, the Igusa quartic { F ( Y , ..., Y ) = 0 } ⊂ P (1 ) above coincides withthe closure of Φ( { W = t W } ) .To describe further relations of the arithmetic quotients to M in (2.4), wenote an isomorphism D K ≃ H of the two domains (see [21, Sect.1.3]). Here wealso note the isomorphism G (2) ≃ Γ M (1 + i ) [21, Prop.1.5.1]. Due to the formerisomorphism, we have the period map P : M → H as a multi-valued map on M with its monodromy group G (2) . Proposition 2.2 ( [21, Thm.4.4.1]) . We have the following commutative diagram: M Y " " ❊❊❊❊❊❊❊❊ P / / H } } ④④④④④④④④ P , where Φ Y defined by A [ Y ( A ) , ..., Y ( A ) , Y ( A ) , ..., Y ( A )] with the semi-invariantsof × matrices given in Appendix D.1. The map Φ Y : M → P ⊂ P is whose branch locus is the Igusa quartic { F ( Y ) = 0 } in P (1 ) ≃ Γ T (1 + i ) \ H . On the other hand, we have noted thatthere is a natural map Γ M (1 + i ) \ H → Γ T (1 + i ) \ H which is generically . Allthese facts are unified by the existence of new theta function Θ which is modularof weight 4 on Γ T [21, Lem.3.1.3], and which vanishes on { W = t W } . Proofs ofthe following results can be found in Proposition 3.1.5 and Theorem 3.2.4 in [21]. Proposition 2.3.
The theta functions satisfy (2.5) Θ( W ) = 3 · (cid:26)(cid:18) X i =1 Θ i ( W ) (cid:19) − X i =1 Θ i ( W ) (cid:27) and this describes the Baily-Borel-Satake compactification of Γ M (1 + i ) \ H as adouble cover of Γ T (1 + i ) \ H ≃ P . This compactification is isomorphic to theGIT compactification M . he geometry of the double cover (2.4), or (2.5), is a well-studied subject in manyrespects. For example, it is known that the double cover is singular along 15 lineswhich are identified with the one dimensional boundary component of the Baily-Borel-Satake compactification. It is also singular at 15 points, which are given asintersections of the lines, representing the zero dimensional components of the Baily-Borel-Satake compactification. In Section 6, we will describe the configuration ofthese singularities, and will find a good resolution from the viewpoint of mirrorsymmetry. Our resolution is also important for introducing the functions λ K which are the mirror maps of our family. (2.3.d) Birational toric variety M , . The Aomoto-Gel’fand system E (3 , should be considered as a hypergeometric system defined over the GIT compacti-fication (or the Baily-Borel-Satake compactification) M . In the next section, wewill find that there appears another variety M , , which is a toric variety, from theanalysis of period integrals. Classically, M , comes from the following birationalcorrespondence to M [24]. Let us consider the six lines { ℓ i } in general positionand select three lines ℓ i , ℓ i , ℓ i to have the map(2.6) { ℓ i } 7→ { ℓ i ∩ ℓ i , ℓ i ∩ ℓ i , ℓ i ∩ ℓ i , ℓ i , ℓ i , ℓ i } , which gives a configuration of three points in P and three lines in P . This definesa rational map from M to the moduli space of configurations of three points andthree lines in P . The variety M , is the GIT compactification of these configura-tions, which turns out to be the following toric hypersurface; M , ≃ { X X X = X X X } ⊂ P . Since three points in general position determines three lines passing through them,given a configuration of three points and three lines in general position, we have sixlines in general position in P . Hence the map (2.6) gives a birational map between M and M , . See Appendix D for its explicit form. This toric variety M , willplay a key role in our analysis of period integrals defined on M .2.4. Toroidal compactification M SecP of P (3 , . In this section, we shall applythe techniques in [17] to give a toric compactification of P (3 , . This is essentialfor describing mirror symmetry of the double cover family of K3 surfaces. The com-pactification M of P (3 , deals with the GL (3 , C ) action on the affine coordinatesof A = ( a ij ) in terms of classical invariant theory. Similarly for the birational toricvariety M , . Our third compactification M SecP arises from reducing the groupactions of GL (3 , C ) and ( C ∗ ) on A ∈ M o , to the diagonal torus actions. (2.4.a) Partial ’gauge’ fixing to T ≃ ( C ∗ ) . To reduce GL (3 , C ) action to thediagonal torus actions, we transform the general matrix A ∈ M o , to the form,(2.7) a b c a b c a b c =: ( E a b c ) . Clearly this reduces the GL (3 , C ) action from the left to the diagonal tori. We notethat there are still residual group actions of the diagonal tori ( C ∗ ) ⊂ GL (3 , C ) combined with the ( C ∗ ) action from the right, i.e., T := n ( g, t ) ∈ GL (3 , C ) × ( C ∗ ) | g (cid:16) E ∗ ∗ ∗∗ ∗ ∗∗ ∗ ∗ (cid:17) t = (cid:16) E ∗ ∗ ∗∗ ∗ ∗∗ ∗ ∗ (cid:17)o (cid:14) ∼ , here ( λg, λ − t ) ∼ ( g, t ) with λ ∈ C ∗ . It is easy to see that T ≃ ( C ∗ ) . We denoteby M E , the subset of M o , consisting of matrices of the form (2.7). We regard M E , as a subset of the 9-dimensional affine C -space A = C . Note that M E , is an opendense subset in A , and the T action naturally extends to A . It is easy to readoff the weights of the T ≃ ( C ∗ ) actions on A . To do that we fix the isomorphism T ≃ ( C ∗ ) , and present the weights of the T -actions on ( a b c ) ∈ A in the followingtable:(2.8) a a a b b b c c c − − − − The toroidal compactification M SecP of P (3 , will turn out to be a toric varietycompactifying the quotient A /T . (2.4.b) Toroidal compactification via the secondary fan. As it will becomeclear when we describe the differential equations of period integrals, the toric varietyof the quotient A /T is given by the data of nine integral vectors which we readfrom the nine column integral vectors in the table (2.8). Following the convention in[10], reordering the columns slightly, we define a finite set A of the integral vectorsby(2.9) A := ( ! , ! , ! , − ! , − ! , − ! , − ! , ! , !) . This set A is a finite set in Z ≡ N . We denote by M the dual of N with the dualpairing h , i : M × N → Z . Proposition 2.4.
The cone
Cone ( A ) generated by A is a Gorenstein cone in N R ,and satisfies Cone ( A ) ∩ { x | h m, x i = 1 } = A with m = (1 , , , , . Proof.
This can be verified by direct computations. (cid:3)
We consider the regular triangulations of the convex hull
Conv ( A ) . Following[11], we have the so-called secondary polytope of A , which we denote by Sec ( A ) .See Appendix A. The secondary polytope is a lattice polytope in L R := L ⊗ R with(2.10) L := Ker (cid:8) ϕ A : Z A → Z (cid:9) , where ϕ A is the integral linear map defined by the × matrix obtained from A in(2.9). The normal fan of Sec ( A ) , called secondary fan, will be denoted by Sec Σ( A ) .The projective toric variety P Sec ( A ) for the polytope Sec ( A ) in L R = L ⊗ R is thetoric variety giving a natural compactification of the quotient A /T . We shalldenote this compactification by M SecP . Proposition 2.5.
The secondary polytope
Sec ( A ) ⊂ L R has 108 vertices. Exceptfor six vertices, the cones from the vertices are regular cones which define smooth ffine charts (coordinate rings) of M SecP . The affine charts corresponding to the6 vertices are singular at the origin and are isomorphic to M LocSecP = Spec C [ C NE ∩ L ] ≃ (cid:26) ( x ij ) ∈ A × | rk (cid:18) x x x x x x (cid:19) ≤ (cid:27) , where C NE ⊂ L R is the cone defined by C NE = Cone ( − , , , , , , , , − , ( 0 , − , , , − , , , , , ( 0 , , − , , , , − , , , ( − , , , , , − , , , , − , , , , , , − , , , − , − , , , , , . Proof.
We can verify the claimed properties directly calculating the secondary poly-tope. The cone C NE is described in Appendix A. (cid:3) Remark . One can also find more details about the combinatorics of the sec-ondary fan in [25] .In the next section, we will observe that the convex hull of the 6 vertices coincideswith a polytope which gives M , , and that M SecP gives a partial resolution ofthe singularities of M , . This observation is the starting point of our analysis of E (3 , defined on M .Explicit forms of hypergeometric series of type E (3 , α , ..., α ) for generalexponents α i are considered in [25] by studying the combinatorial aspect of thesecondary polytope Sec ( A ) . However, it should be noted that our system E (3 , has special values of exponents α = · · · = α = , which belongs to the casescalled resonant , and is beyond the consideration in [25]. In fact, we need to findout detailed relationships between the moduli spaces M , M , and M SecP to writethe solutions for this case. After formulating the relationships, we will observe inSection 7 that the techniques in [16, 17] developed in mirror symmetry and theresults in [22, 21] merge quite nicely in a general framework, i.e., D -module onGrassmannians [1, 2, 8] . GKZ hypergeometric system from E (3 , It is known in general that the Aomoto-Gel’fand system on Grassmannians isexpressed by the Gel’fand-Geraev and Gel’fand-Kapranov-Zelevinski system (GKZsystem for short) when we reduce the GL ( n, C ) -action to tori by making a “partialgauge” of the form (2.7) (see [2, Sect.3.3.4]). Here we study the period integral(2.1) with the reduced form (2.7) to set up the GKZ system.3.1. GKZ hypergeometric system from E (3 , . Let us take the parameters inthe six lines ℓ i as in (2.7). Then we can write the holomorphic two form as(3.1) dµ p Π i =1 ℓ i = d x ∧ d y p xy ( a + a x + a y )( b + b x + b y )( c + c x + c y )= 1 q(cid:0) a + a x + a yx (cid:1)(cid:0) b + b y + b xy (cid:1)(cid:0) c + c x + c y (cid:1) d x ∧ d yxy , here we take the affine coordinate z = 1 of P . We observe that the finite set A in(2.9) can be interpreted as the exponents of the three Laurent polynomial factorsin the denominator, if we write A as follows: A = (cid:26) e × (cid:18) (cid:19) , e × (cid:18) (cid:19) , e × (cid:18) (cid:19) ,e × (cid:18) − (cid:19) , e × (cid:18) − (cid:19) , e × (cid:18) − (cid:19) , e × (cid:18) − (cid:19) , e × (cid:18) (cid:19) , e × (cid:18) (cid:19) (cid:27) , where we e , e , e are the basis of the first factor in Z = Z × Z . Let us writethe three Laurent polynomial factors as f ( a, x , y ) , f ( b, x , y ) , f ( c, x , y ) so that (3.1)becomes p f ( a, x , y ) f ( b, x , y ) f ( c, x , y ) d x ∧ d yxy . Observe the striking similarity with the corresponding forms we encountered in afolklore paper [16], except the appearance of the square root in the denominator.
Proposition 3.1.
Let A be as given in (2.9). The period integral (2.1) with itsintegrand (3.1) satisfies GKZ A -hypergeometric system [10] with exponents β = t ( − , − , − , , . Proof.
This follows easily by looking at invariance properties under the torus action T of the period integral, see [16, 17]. The only difference from there is in theexponent β , which is explained by the square root in the denominator. We leavethe derivation as an easy exercise for the reader. (cid:3) Remark . From the first line to the second line of (3.1), the division by xy hasbeen made by making a choice which factor of xy goes to which factor of the threeparentheses. There are six combinatorially different ways in total. Recall that wehave chosen the isomorphism T ≃ ( C ∗ ) for the weights (2.8) so that the resultingset A is compatible with the choice made in (3.1). We will return to this point inthe next subsection.3.2. Boundary points (LCSLs) of the GKZ system.
A fundamental objectin mirror symmetry is a special boundary point in the moduli space of Calabi-Yau manifolds, called a LCSL, which appears as the intersection of certain normalcrossing boundary divisors of suitable compactification of the moduli space. In thecase of Calabi-Yau complete intersections in toric varieties, it is well known thatsuch compactifications are naturally obtained by finding a suitable toric resolutionof the compactification M SecP [16, 17]. (3.2.a) Resolutions of M SecP . Under the identification L ≡ Z in (B.1), we have(3.2) C NE = Cone (cid:26) (1 , , , , (0 , , − , , (1 , − , , , (0 , , , , , , − , ( − , , , (cid:27) , for the cone C NE ⊂ L R . Lemma 3.3. (1) The dual cone C ∨ NE is generated by ρ , · · · , ρ where ρ = (1 , , , , ρ = (0 , , , , ρ = (0 , , , ,ρ = (1 , , , , ρ = (0 , , , .
2) Without adding extra ray generators, there are two possible decompositions of C ∨ NE , namely, (3.3) C ∨ NE = σ (1)1 ∪ σ (1)2 = σ (2)1 ∪ σ (2)2 ∪ σ (2)3 with σ (1) i = Cone { ρ , ρ , ρ , ρ − i } ( i = 1 , and σ (2) i = Cone { ρ j , ρ k , ρ , ρ } ( { i, j, k } = { , , } ) . (3) All σ ( k ) i are smooth simplicial cones, and hence each in (2) defines a resolutionof the singularity at the origin of Spec C [ C NE ∩ L ] . The first and the second decom-positions in (2) correspond, respectively, to the left and the right resolutions shownFig.1.Proof. All the claims can be verified by explicit calculations. (cid:3)
Proposition 3.4.
Choose a subdivision of (3.3), independently, at each of the sixaffine charts of M SecP corresponding to the six singular vertices in Proposition2.5. For each choice of the subdivisions, we have a resolution of M SecP , and thedifference of the choice in (3.3) is represented by four dimensional flip shown inFig. 1.Proof.
Our proof is based on the explicit construction of the secondary fan
Sec Σ( A ) ,which consists of 108 four dimensional cones. Since all cones except the six aresmooth, we obtain a resolution by choosing a subdivision for each of the six conesas claimed. The four dimensional flip should be clear in the form of the singularityexpressed by the rank condition in Proposition 2.5. (cid:3) We shall write f M SecP and f M + SecP , respectively, for the resolution where all sixlocal resolutions are of the left type and the right type in Fig.1.
Fig.1
Four dimensional flip in the resolutions of
Spec C [ C NE ∩ L ] .All boundary points o ( a ) i are LCSLs. (3.2.b) Power series solutions and Picard-Fuchs equations. In this subsec-tion, we give the power series solutions of the GKZ A -hypergeometric system near he LCSL in the the affine chart Spec C [ C NE ∩ L ] (see Appendix A). To simplifythe form of the power series, we normalize the period integral (2.1) as follows:(3.4) ω C ( a ) := p a b c ¯ ω C ( a ) . Definition 3.5.
Let ( σ ( k ) i ) ∨ be the dual cone of σ ( k ) i in (3.3), which is smooth. Werepresent ( σ ( k ) i ) ∨ in L R by using (B.1). Then in terms of its primitive generators,we have ( σ ( k ) i ) ∨ = Cone n ℓ (1) , ℓ (2) , ℓ (3) , ℓ (4) o . Let z m := a ℓ ( m ) = Q i =1 a ℓ ( m ) i i be the affine coordinates on Spec C [( σ ( m ) i ) ∨ ∩ L ] witharranging the parameters a := ( − a , − b , − c , a , a , b , b , c , c ) . Then the hypergeometric series associated to σ ( a ) i is defined to be(3.5) ω ( z ) = X n ,...,n ≥ ) Q i =1 Γ( n · ℓ i + ) Q i =4 Γ( n · ℓ i + 1) z n z n z n z n , where n · ℓ := P k n k ℓ ( k ) (see [16, 17]).The hypergeometric series w ( z ) is the unique power series solution of the GKZ A -hypergeometric system on M SecP near a LCSL point. We now use the methoddeveloped in [17] to determine the complete set of the Picard-Fuchs differentialoperators. To show the calculations, we take the affine chart
Spec C [( σ (1)1 ) ∨ ∩ L ] as an example. It should be clear that the constructions below are parallel for theother cases Spec C [( σ ( k ) i ) ∨ ∩ L ] .As the primitive generator of ( σ (1)1 ) ∨ ⊂ L R , we first obtain ℓ (1) = ( − , , , , , , , , − ,ℓ (2) = ( 0 , − , , , − , , , , ,ℓ (3) = ( 0 , , − , , , , − , , ,ℓ (4) = ( 0 , , , − , , − , , − , . The power series (3.5) now becomes(3.6) ω ( z ) = X n ,n ,n ,n ≥ c ( n , n , n , n ) z n z n z n z n with the coefficients c ( n ) = c ( n , n , n , n ) given by c ( n ) := 1Γ( ) Γ( n + )Γ( n + )Γ( n + )Π i =1 Γ( n − n i + 1) · Π ≤ j Spec C [( σ (1)1 ) ∨ ∩ L ] . Proposition 3.6. The period integral ω ( z ) in (3.6) is the only power series so-lution near a LCSL given by the origin of Spec C [( σ (1)1 ) ∨ ∩ L ] ≃ C . The originis the special point (LCSL) where all other linearly independent solutions containsome powers of log z i ( i = 1 , ..., . Proof. The first claim can be verified by the set of differential operators in AppendixC. For the second claim, we will find a closed formula for the logarithmic solutions.The closed formula will be described in detail in [18]. (cid:3) Calculations are completely parallel for all other origins o ( k ) i of the affine charts Spec C [( σ ( k ) i ) ∨ ∩ L ] of the resolutions. One can verify the corresponding propertiesin the above proposition hold for all o ( k ) i . Remark . As noted in Remark 3.2, the six singular vertices in the secondarypolytope Sec ( A ) come from the combinatorial symmetry when reading A from theperiod integral (3.1). Hence, up to permutations among the variables a i , b j and c k , respectively, the hypergeometric series which we define for each of the six affinechart have the same form as (3.6). Therefore the Picard-Fuchs differential operatorshave the same form, up to suitable conjugations by monomial factors, for all sixaffine charts of the form Spec C [ C NE ∩ L ] from the vertices T , ..., T . Based on thissimple property, we will have the same Fourier expansions for the certain lambdafunctions when expanded around the boundary points. Details are described in[18]. 4. M , from period integrals As presented in [16, 17] for the case of Calabi-Yau complete intersections intoric varieties, GKZ hypergeometric systems provide powerful means for calculatingvarious predictions of mirror symmetry. One may naively expects that this is alsothe case for E (3 , . However, it turns out that we need to further understandrelationships between the compactifications M SecP , M , and finally M . In thissection, we will find that the compactification M , arises naturally from evaluatingperiod integrals. We will see that M SecP is actually a partial resolution of M , . .1. Power series from residue calculations. Recall that, when determiningPicard-Fuchs differential operators in the previous section, we have normalized theperiod integral (2.1) by ω C ( a ) = √ a b c ¯ ω C ( a ) . Under this normalization, bymaking use of the expansion √ P = P r n P n , we can evaluate the period integralover the torus cycle γ = {| x | = | y | = ε } as follows ˆ √ a b c q ( a + a x + a yx )( b + b y + b xy )( c + c x + c y ) d x d yxy = ˆ X n,m,k r n (cid:18) a a x + a a yx (cid:19) n r m (cid:18) b b y + b b xy (cid:19) m r k (cid:18) c c x + c c y (cid:19) k d x d yxy by formally evaluating the residues. Lemma 4.1. We have the period integrals over the torus cycle γ as a power seriesof (4.1) x := a c a c , y := a b a b , z := b c b c , u := − a b c a b c , v := − a b c a b c which satisfy the equation xyz = uv . Eliminating the powers of v , the result isformally expressed by (4.2) ω ( x, y, z, u ) := ∞ X l = −∞ X m,n,k ≥ max { , − l } c ( n, m, k, l ) x n y m z k u l , where c ( n, m, k, l ) := 1Γ( ) Γ( m + n + l + )Γ( n + k + l + )Γ( m + k + l + ) m ! n ! k ! ( m + l )! ( n + l )! ( k + l )! . Proof. The evaluation of the residues is straightforward (cf. [3, 17]). The closedformula of the coefficients c ( n, m, k, l ) can be deduced from the formal solutions ofthe GKZ system [10]. (cid:3) Proposition 4.2. The Laurent series ω ( x, y, z, u ) defines a regular solutions arounda point [0 , , , , , ∈ M , under the following identification of the parameters x, y, z, u, v with the affine coordinate of M , : (cid:8) [ x, y, z, u, v, ∈ P | xyz = 1 uv (cid:9) ⊂ M , . Proof. By the definitions of x, y, z, u, v , we have the relation xyz = uv . The claimis clear since we have a power series of x, y, z, u, v in Lemma 4.1 (before eliminating v ). (cid:3) Remark . Recall that we have made a choice, among six combinatorial possibil-ities, from the first line to the second line of (3.1) as noted in Remark 3.2. It iseasy to deduce that, if we change our choice there, we will have the same powerseries but with different variables, which corresponds to expansions around differentcoordinate points of P (cf. Remark 3.7). Namely, when we reduce the GL (3 , C ) symmetry to the diagonal tori as in (2.7), we may consider that the period integral(3.1) is defined on M , = { X X X = X X X } ⊂ P . .2. M SecP and M , . We have seen in Proposition 3.6 that the special boundarypoints (LCSLs) appear in the resolutions of M SecP . Here it turns out that M SecP gives a partial resolution of M , . Proposition 4.4. The toric hypersurface M , ⊂ P contains all coordinate linesof P . The singularities of M , consist of six coordinate points p i ( i = 0 , .., of P and nine coordinate lines p i p j (0 ≤ i ≤ , ≤ j ≤ .Proof. Since all claims are easy to verify from the defining equation of the hyper-surface, we omit the proofs. (cid:3) Fig.2 Singularities of M , . Solid lines represent the coordinatelines p i p j (0 ≤ i ≤ , ≤ j ≤ along which M , is singular.Broken lines are the other coordinate lines contained in M , .The following lemma is our first step to relate M SecP and M , . To state it, werecall that the the secondary polytope Sec ( A ) has 108 vertices, whose associatedcones define coordinate rings of the affine charts of M SecP . Of the 108 vertices,the six vertices V given in Appendix A are singular while the rest are smooth (seeProposition 2.5). Lemma 4.5. We have M , = P Conv ( V ) .Proof. This follows from the explicit calculation of Conv ( A ) . We list the six vertices V of Sec ( A ) in Appendix A. From the list, it is straightforward to see the claim. (cid:3) By the obvious symmetry of M , , we may restrict our attention to the localaffine geometry M Loc , := { xyz = uv } ⊂ C , and deduce its relation to the resolution f M SecP . If we read the exponents of thevariables in (4.1), we can write the toric singularity M Loc , using the lattice (2.10)as M Loc , = Spec C [ C ∩ L ] , here(4.3) C := Cone ( − , , − , , , , , , , ( − , − , , , , , , , , ( 0 , − , − , , , , , , , ( − , − , − , , , , , , , ( − , − , − , , , , , , . Note that the five generators ℓ ∈ L of C listed here express the the affine coordi-nates x, y, z, u, v in (4.1) by the monomials a ℓ . Under (B.1), we can also write C by C = Cone { (0 , , , , (1 , , , , (0 , , , , (1 , , , , (0 , , , }} ⊂ R . Note, from the form of C in (4.3) and C NE in Appendix A, that C and C NE arecones from the same vertex T of Sec ( A ) . Lemma 4.6. We have C ⊂ C NE ⊂ L R for the cone C NE .Proof. Since the vertex is chosen in common for C NE and C , the claimed inclusionis easy to verify. (cid:3) In Appendix A, we have listed the primitive generators of the dual cone C ∨ ,which we denote by µ , · · · , µ in order. Similarly we write the primitive generatorsof the dual cone C ∨ NE by ρ , ..., ρ . Note that, by Lemma 4.6, we have the reversedinclusion as a set for the dual cones, i.e., supp C ∨ ⊃ supp C ∨ NE holds for the supports, in particular, the rays generated by ρ , · · · , ρ are containedin C ∨ . Recall that the dual cone C ∨ NE has two possible subdivisions into smoothsimplicial cones as described in Lemma 3.3 (2). In the following lemma, we considersubdivisions of the dual cone C ∨ using all rays generated by µ , · · · , µ , ρ , · · · , ρ . Lemma 4.7. Up to the subdivisions of C ∨ NE in Lemma 3.3 (2), there is a uniquesubdivision of C ∨ into smooth simplicial cones which contains the dual cone C ∨ NE as a simplicial subset.Proof. By explicit construction of all possible subdivisions, via a C++ code TOP-COM [23], we find 54 subdivisions. We verify the claimed property from them. (cid:3) Lemma 4.8. By the unique subdivision of C ∨ in Lemma 4.7 which contains C ∨ NE as the simplicial subset, we have a partial resolution of the singularity M Loc , =Spec C [ C ∩ L ] .Proof. The claim is clear, since C ∨ consists of smooth cones up to subdivisions of C ∨ NE . (cid:3) Proposition 4.9. The partial resolutions at each singular points gives globally apartial resolution M SecP → M , . Proof. Our proof is based on the explicit coordinate description of M SecP calcu-lating the secondary polytope. See also Remark 4.10 below. (cid:3) emark . Toric resolutions of M SecP have been described in Proposition 5.7.In the next section, we will obtain the same resolutions by blowing-up along thesingular locus of M , (Proposition 5.7). In Fig.4, we depict one of the two possibleresolutions of M Loc , schematically. As we see from the picture, the resolution ofthe singularity is covered by 19 affine coordinate charts which correspond to 19maximal dimensional cones in the subdivision of C ∨ . If we remove the subdivisionof C ∨ NE ⊂ C ∨ , then the number reduces to 18, which is explained by 17 smoothmaximal cones and one singular cone C ∨ NE corresponding to Spec C [ C NE ∩ L ] . Onecan also see the claim in Proposition 4.9 in a simple counting × (seeProposition 2.5). 5. More on the resolutions of M , In this section, we will describe the resolution without recourse to the toricgeometry of the secondary fan. This will allow us to relate M SecP to the geometryof the Baily-Borel-Satake compactification M . Recall that we have defined M Loc , = Spec C [ C ∩ L ] ≃ { ( x, y, z, u, v ) | xyz = uv } , which which describes the local structure of the singularities in M , . We shallwrite X = M Loc , for short in what follows.5.1. Blowing-up X ′ → X along the singular locus. From the defining equation xyz = uv , it is easy to see that the affine hypersurface X ⊂ C is singular alongthe three coordinate lines of x, y, z coordinates (cf. Subsection 4.2). Note that wecan write the union of these lines in C by Γ := { u = v = xy = yz = zx = 0 } . We will consider the blow-up π : X ′ → X along this locus Γ . Let us first introducethe blow up f C ⊂ C × P starting with the relations u : v : yz : zx : xy = U : V : W : W : W , for ( u, v, x, y, z ) × [ U, V, W , W , W ] ∈ C × P . The ideal I f C of the blow-up f C ⊂ C × P is an irreducible component of the scheme defined by the aboverelations. We denote by π : f C → C the natural projection. Then the blow-up X ′ is the strict transform of X ⊂ C by the birational map π . Proposition 5.1. The blow-up X ′ is given in C × P by the following equations: (5.1) W W = U V z, W W = U V y, W W = U V x,W x = U v W y = U v W z = U vW x = V u, W y = V u, W z = V u and (5.2) W u = yzU, W u = zxU, W u = xyU,W v = yzV, W v = zxV, W v = xyV. roof. The ideal I f C and the equation xyz = uv define the ideal I T of the totaltransform of X . Calculating the primary decomposition of I T by Singular [5], wesee that the claimed equations generate the ideal of X ′ . (cid:3) Fig.3 Exceptional divisors E x , E y and E z in the blow-up X ′ . Theirjunction locus is scaled up in the right figure. Proposition 5.2. The blow-up π : X ′ → X has the following properties: (1) The π -exceptional divisor has three irreducible components; one for eachcoordinate line of x, y, z coordinates. We call the irreducible components E x , E y , E z , respectively.(2) The components E x , E y , E z have fibrations over the corresponding coor-dinate lines. The π -fiber over a point p ∈ Γ is ( P ) if p is not the origin o , while over the origin it is the union of three copies P i ( i = 1 , , of P which are glued along one line ℓ := P (see Fig.3). Over the origin, thecomponents E x , E y , E z glue together by the following relations: E x | π − ( o ) = P ∪ P , E y | π − ( o ) = P ∪ P , E z | π − ( o ) = P ∪ P . (3) The blow-up X ′ is singular only at two isolated points, say, p and p on ℓ .The singularities at these points are isomorphic to the affine cone over theSegre ( P ) .(4) The components E x , E y and E z are singular only at p and p with ODPs. Proof. The claimed properties follow from the equations in Proposition 5.1. For(1) and (2), because of the obvious symmetry, we only need to consider the case of x -axes. Set y = z = u = v = 0 in (5.1) assuming x = 0 . Then we obtain W = 0 and W W = U V x , from which we see π − ( p ) ≃ P × P ( p = o ) as claimed. When x = y = z = u = v = 0 , the equations (5.1) become W W = W W = W W = 0 ,from which we obtain π − ( o ) = { o × [ U, V, W , , } ∪ { o × [ U, V, , W , } ∪ { o × [ U, V, , , W ] } =: P ∪ P ∪ P . lso we see that P ∩ P ∩ P = { o × [ U, V, , , } =: ℓ as claimed. It is easy to seethe claimed forms of E x | π − ( o ) , E y | π − ( o ) and E x | π − ( o ) .To show (3), we express X ′ in affine coordinates. By obvious symmetry, we onlyhave to consider X ′ | W =0 and X ′ | U =0 . Let us first describe the restriction X ′ | W =0 by setting W = 1 . Then we obtain the relations W = U V z, W = U V y, x = V u from (5.1) and also u = U yz, v = V yz from (5.2). From these relations, we seethat X ′ | W =0 is isomorphic to C with the coordinates y, z, U, V . By symmetry,similar results hold for other cases W = 0 and W = 0 . In particular, X ′ | W i =0 aresmooth for i = 1 , , . Next, let us describe X ′ | U =0 by setting U = 1 . From (5.1), we obtain(5.3) W W = V z, W W = V y, W W = V x,W x = V u, W y = V u, W z = V u in addition to v = V u which eliminates v . Also from (5.2), we have(5.4) W u = yz, W u = zx, W u = xy and also W v = yzV, W v = zxV, W v = xyV , where the latter three relationsare consequences other relations. We note that the equations (5.3) and (5.4) aredeterminants of × sub-matrices of the 2-hypermatrix given in the equation below.Moreover, the relations (5.3) and (5.4) are solved by a ijk written in terms of thehomogeneous coordinates ([ a , a ] , [ b , b ] , [ c , c ]) ∈ ( P ) ; a b c a b c a b c a b c a b c a b c a b c a b c (cid:0) a ijk (cid:1) = = V W W zW xy u ⑧⑧⑧ ⑧⑧⑧⑧⑧⑧ ⑧⑧⑧ ⑧⑧⑧ ⑧⑧⑧⑧⑧⑧ ⑧⑧⑧⑧ Thus we see that the relations (5.3) and (5.4) define the affine cone of the Segre ( P ) in C with the affine coordinates x, y, z, u, V, W , W , W , which is singular at thevertex (the origin of C ). Note that the vertex corresponds to the point p := o × [1 , , , , ∈ X ′ which is on the line ℓ = { o × [ U, V, , , } . By symmetry, theother case X ′ | V =0 can be described similarly with the vertex p := o × [0 , , , , on the line ℓ .The claim (4) follows from the proof for (3). For example, we set y = z = u = v = 0 in the equations (5.3) and (5.4). Then we can verify the claimed propertyfor E x . (cid:3) Note that p and p are the only singular points of X ′ . Let π : e X → X ′ bethe blow-up at p and p . We denote by e E x , e E y , e E z the strict transforms of the π -exceptional divisors E x , E y , E z respectively. Proposition 5.3. The blow-up π : e X → X ′ introduces exceptional divisors D p , D p which are isomorphic to ( P ) . The resulting composite of the blow-ups of X gives resolution of singularities π ◦ π : e X → X . Moreover, the union e E x ∪ e E y ∪ e E z ∪ D p ∪ D p is a simple normal crossing divisor.Proof. The first two claims follow from Proposition 5.2. The last assertion alsofollows from the explicit computations. (cid:3) Fig.4 The blow-up of X ′ at p , p in the junction. The intersectionpoints o (1) k = f E x ∩ f E y ∩ f E z ∩ D p k ( k = 1 , and e ℓ can be identifiedin the left of Fig.1. Remark . As shown in Fig.4, the strict transforms of the three P i ( i = 1 , , under the blow-up π : e X → X ′ are P blown up at two points.Making the blow-up e X → X = M Loc , at each singular points of M , , we obtainthe resolution f M SecP of the partial resolution M SecP → M , in Proposition 3.4.Note that, in the resolution f M SecP thus obtained, we have the resolution e X (theleft in Fig.1) at all six singular points.5.2. Flipping the line ℓ in e X to P . Recall that we have introduced the line ℓ = P ∩ P ∩ P in X . Correspondingly, we have e ℓ = e P ∩ e P ∩ e P on e X . Here andin what follows we put e to indicate the the strict transform of a subvariety of X ′ .We can also write ℓ = E x ∩ E y ∩ E z and e ℓ = e E x ∩ e E y ∩ e E z by Proposition 5.2. Let N e ℓ/ e X be the normal bundle of e ℓ in e X . Lemma 5.5. We have N e ℓ/ e X ≃ O P ( − ⊕ .Proof. By Proposition 5.3, e E x , e E y and e E z are smooth on e X . Since e ℓ = e E x ∩ e E y ∩ e E z ,we have only to show that e E x · e ℓ = e E y · e ℓ = e E z · e ℓ = − . By symmetry, itsuffices to show that e E x · e ℓ = − . Since e E x ∩ e E y = e P and e P ∩ e P = e ℓ , we have e E x · e ℓ = ( e P · e ℓ ) e E y = ( e ℓ ) e P . Note that e P i ( i = 1 , , is a P blown-up at two pointsand e ℓ is a ( − -curve on e P i . Therefore ( e ℓ ) e P = − as claimed. (cid:3) Proposition 5.6. There is a flip which transforms the line e ℓ to P . roof. Here we only consider analytically for simplicity. See the proof of Theorem6.12 for an algebraic construction of the flip. Since N e ℓ/ e X ≃ O P ( − ⊕ , by blowing-up along the line ℓ , we obtain ℓ × P as the exceptional divisor. Contracting thisto P , we obtain the flip (cf. Fig.1). (cid:3) We denote by e X + → X the resulting resolution after the flip of the resolution e X → X = M Loc , . Proposition 5.7. Making resolutions e X → X or e X + → X locally at each of sixisomorphic singular points of M , , we obtain the same resolution as the toricresolutions of M , in Proposition 4.9 and Proposition 3.4.Proof. We verify the claim explicitly by writing the resolutions of M SecP in Propo-sition 3.4. Here we only sketch our calculations. As described in the proof ofProposition 3.4, the partial resolution M SecP of M , is covered by 108 affinecharts, among which six charts are singular. The singular charts are isomorphic to Spec C [ C NE ∩ L ] which has two resolutions shown in Fig.1. By explicit calculations,we find that 108 affine charts are grouped into six isomorphic blocks of 18 charts(one singular and 17 smooth charts). We verify that each block is isomorphic to e X or e X + after making a resolution of the singular chart. (cid:3) The above proposition provides us a global picture of the parameter space of theGKZ A -hypergeometric system in Proposition 3.1. Our task in the next section isto make a covering of the parameter space E (3 , by certain Zariski open subsetsof the parameter space of the GKZ A -hypergeometric system. Remark . Instead of constructing the resolution e X → X starting with the blow-up X ′ along Γ , we can also make a resolution by first blowing-up along z -coordinateline and then blowing-up along x - and y -coordinate lines. Since the (strict trans-forms of) x - and y -coordinate lines are separated by the first blowing-up along z -coordinate line, and the singularities along these lines are of A type, we obtaina resolution b X → X in this way. Note that the resolution b X → X introduces onlythree exceptional divisors from the blowing-ups, and hence this is not isomorphicto e X → X in Proposition 5.3 nor e X + → X . Moreover, the generalized Frobeniusmethod developed in [17, 16] does not apply to the resolution b X → X . Recall thatthe generalized Frobenius method provides a closed formula for the local solutionsaround special boundary points (LCSLs), such as o ( k ) i in e X → X or e X + → X , givenby normal crossing boundary or exceptional divisors. In the resolution b X → X ,there is no way to have such special boundary points by the three exceptionaldivisors.6. Blowing-up the Baily-Borel-Satake compactification M We will study the relationship between the Baily-Borel-Satake compactification M and the compactification M , , which appears naturally from computing theperiod integrals. We recall that the compactification M is birational to M , withthe birational map given by (2.6). .1. Birational map φ : M , M . Since both M and M , have descrip-tions in term of GIT quotients, the birational map φ can be given explicitly bywriting the relevant semi-invariants [6, 24]. We have sketched the results in Ap-pendix D; in particular, we have given the explicit form of the birational mapusing the (weighted-)homogeneous coordinates [ X , X , ..., X ] ∈ P for M , and [ Y , ..., Y , Y ] ∈ P (1 , for M . Lemma 6.1. The following properties holds: (1) φ defines a map φ : M , \ { [1 , , , , } → M , and(2) φ − defines a map φ − : M \ { Y = 0 } → M , . Proposition 6.2. Define the following divisor in M , : (6.1) D = { X + X + X − X − X − X = 0 } . Then the birational map φ restricts to a 1 to 1 map (6.2) φ : M , \ D → φ ( M , \ D ) ⊂ M to its image in M . The proofs of the above lemma and proposition are easy from the explicit forms(D.4) and (D.5) of the birational maps φ and φ − , respectively. Further properties,e.g., the restriction φ : D \ { [1 , , , , } → M , can be worked out, but we leavethese to the reader (see [24, Sect.2.4]).6.2. Singularities of M . Singularities of M are well-studied objects in the liter-atures (see [22, 19] for example). Here we summarize the results from our viewpointsand using the (weighted-)homogeneous coordinate [ Y , Y , ..., Y ] of P (1 , . Proposition 6.3. The variety M is singular along 15 lines which intersect at 15points which, respectively, correspond to one dimensional boundary components andzero dimensional boundary components in the Baily-Borel-Satake compactification.These 15 lines are located in { Y = 0 } ≃ P ⊂ P (1 , .Proof. The results are well-known in the literatures (see [28, 19] for example). Anexplicit description of the boundary components is given in Appendix E. (cid:3) Proposition 6.4. Each of the 15 points of singularities is given by the intersec-tion of corresponding three lines. Vice versa, each of the 15 lines contains threeintersection points with other two lines at each intersection.Proof. We verify the claimed properties using the equations for the 15 lines inAppendix E and schematic description of the 15 lines in Fig.5. (cid:3) Proposition 6.5. The 9 lines of singularities in M , described in Proposition 4.4correspond to 9 of 15 lines in M by the birational map φ : M , M . Inparticular, the local structure M Loc , near the 6 point is isomorphically mapped tothe corresponding intersection points of lines in M . ig.5 Configuration of the 15 lines. φ ( p k ) represent the images ofthe coordinate points p k in M , . Lines L k ( k = 1 , ..., are givenexplicitly in Appendix E. L , ..., L shown in the left correspondto the 9 singular lines in M , . Lines L , ..., L are in the divisor { Y = 0 } where φ − is not defined, and intersect with L , ..., L atone point as shown in the right. Proof. Recall that the 9 lines in M come from coordinate lines of P and intersectat 6 coordinate points. None of the 9 lines nor their intersection points are containedin D (6.1). Hence these lines determine the corresponding lines in M under thebirational map φ , along which M is singular. Also the local structure M Loc , ismapped isomorphically to M . (cid:3) In the next subsection, we will see that the local structure near all the 15 singularpoints in M are isomorphic to M Loc , .6.3. S action on M . Now recall that the homogeneous coordinate Y i is related tothe × matrix A by (D.1). We note that there is a natural action of the symmetricgroup S sending A → Aσ := Aρ ( σ ) by the permutation matrix ρ ( σ ) representing σ ∈ S . This naturally induces linear actions on homogeneous coordinates Y i ( A ) Y i ( Aσ ) . Lemma 6.6. The action Y i = Y i ( A ) Y i ( Aσ ) is linear and preserves the homo-geneous weights of the coordinate [ Y , ..., Y , Y ] ∈ P (1 , .Proof. The claim is clear since Y i = Y i ( A ) are generators of the semi-invariants of GL (3 , C ) of fixed degrees, and Y i ( Aσ ) are semi-invariants of the same degree with Y i ( A ) . (cid:3) Geometric meaning of the action A → Aσ is simply that it changes the order ofthe (ordered) six points in P . From Lemma 6.6, it is easy to deduce the followingproposition. Proposition 6.7. The linear action Y i = Y i ( A ) Y i ( Aσ ) naturally defines thecorresponding automorphism ψ ( σ ) : M ≃ M for σ ∈ S . We combine this isomorphism with the birational map φ : M , M . efinition 6.8. For σ ∈ S , we define the following composite of ψ ( σ ) and φ : φ σ : M , M ψ ( σ ) ≃ M . Covering M by open sets of toric varieties. We now combine all theresults about the moduli spaces M , and M . We first recall that M is given bya hypersurface in P (1 , . Lemma 6.9. The hypersurface M misses the point [0 , , , , , ∈ P (1 , .Proof. We simply verify the property from the definition (2.4). (cid:3) Lemma 6.10. Take the following permutations σ ∈ S e, (34) , (35) , (24) , (25) and name these by σ k ( k = 0 , , ..., in order. Then under the automorphism ψ ( σ k ) : M ≃ M , the hyperplane { Y = 0 } ⊂ M transforms to { Y k = 0 } ⊂ M for k = 0 , , ..., , respectively.Proof. By lemma 6.6, Y ( Aσ k ) is linear in Y i ’s. We derive the claimed results bycalculating the semi-invariants given in (D.1) under the permutations. (cid:3) Proposition 6.11. The moduli space M is covered by copies of M , \ D . Moreprecisely, we have M = [ k =0 φ σ k ( M , \ D ) . Proof. By Lemma 6.9, one of the homogeneous coordinate Y , ..., Y does not vanishfor any point of M . Then, due to Lemma 6.10, any point is contained in the unionof the isomorphic images φ σ k ( M , \ D ) of M , \ D (see (6.2) and Definition6.8). (cid:3) The local structures near each of the 15 singular points in M is isomorphicto the local structure of M Loc , . Making the resolution e X → X = M Loc , given inProposition 5.3 at each singular point, we have the resolution f M → M . Namely,let f : M ′ → M be the blow-up along Sing M , which is the union of lines.Then, M ′ has × singular points. Let f : f M → M ′ be the blow-up at all thesingular points.Recall that locally we have another resolution e X + . In the following theorem,we can globalize this to another resolution of M connected with f M by a -dimensional flip. Theorem 6.12. There exists another resolution f M +6 of M which is connectedwith f M by a -dimensional flip.Proof. We have already constructed the flip e X e X + of e ℓ locally analyticallyin Proposition 5.6. Let e ℓ , . . . , e ℓ ⊂ f M be the copies of P over the fifteensingular points of M . The remaining problem is to construct the flips of e ℓ , . . . , e ℓ algebraically and globally. The following properties guarantee this. We will provethem in Appendix F.Let E be the f -exceptional divisor and e E its strict transform on f M . Set f := f ◦ f . Then − ( K f M + 1 / e E ) is f -nef, and is numerically f -trivial only for e ℓ , . . . , e ℓ .(2) There exists a small contraction ρ : f M → M over M contracting exactly e ℓ ∪ · · · ∪ e ℓ .(3) The contraction ρ is a log flipping contraction with respect to some klt pair ( f M , D ) .(4) The flip f M f M +6 of ρ exists and it coincides locally with the flipconstructed as in Proposition 5.6.This completes the construction of the resolution. (cid:3) Remark . By Theorem 6.12, we have two algebraic resolutions f M → M and f M +6 → M , which are related by a four dimensional flip. Interestingly, it willturn out in [18] that these two possibilities of algebraic resolutions result in twonon-isomorphic definitions of the lambda functions λ K on M .7. Hypergeometric D -modules on Grassmannians In this section, we combine the results of earlier sections to give our main resultsof this paper.We have obtained a global picture for the moduli space M in terms of thetoric variety M , which is closely related to the toric variety M SecP . With theseresults in hand, we now look at the hypergeometric system E (3 , defined on itsparameter space M . To have a global picture, it is better think of E (3 , as thecorresponding D -module on M . In this language, our first result is Theorem 7.1. On each of the open set φ σ k ( M , \ D ) ( k = 0 , , .., of M , thehypergeometric D -module of E (3 , restricts to the D -module of the GKZ systemin Proposition 3.1.Proof. This follows by combining the results in Sections 3 and 4 with Proposition6.11. (cid:3) The GKZ A -hypergeometric system has the natural compactification M SecP interms of the secondary fan. As we saw in Proposition 3.6, the special boundarypoints (LCSLs) arise in the resolutions of M SecP . By Propositions 4.9 and 5.7,the resolutions of M SecP are in fact the resolutions of M , , and are given bythe resolutions of the local singularity e X → M Loc , . We have transformed theselocal structures to M by the isomorphisms φ k : M , \ D ≃ φ σ k ( M , \ D ) ,and obtained the desired resolutions of M . Among the resolutions, in particular,we have constructed two algebraic resolutions f M and f M +6 . Our second result isabout the LCSLs in these resolutions. Theorem 7.2. In the above resolutions of M , the LCSLs are given by the inter-sections of normal crossing divisors, which are given by isomorphic images under φ σ k ( k = 0 , , ..., of the divisors of the blow-ups e X → X = M Loc , or their flips e X + → X .Proof. The claims are shown in Sections 5 and 6. By Proposition 3.6 and Proposi-tion 5.7, the boundary points are in fact the desired LCSLs. (cid:3) n a companion paper [18], we will construct the so-called mirror maps from thelocal solutions near each LCSL. The mirror maps turn out to be generalizationsof the classical λ -function for the Legendre family of the elliptic curves. We willcall these new examples of mirror maps λ K -functions. Then, Theorem 7.2 impliesthat the λ K -functions have nice q -expansions (Fourier expansions) at the boundarypoints in the suitable resolutions of the Baily-Borel-Satake compactification of thedouble cover family of K3 surfaces. As mentioned in Remark 6.13, it will turn outin [18] that there are two non-isomorphic definitions of λ K -functions correspondingto the two algebraic resolutions f M and f M +6 . Remark . For the double cover family of K3 surfaces, the two basically differentdefinitions of the moduli space are isomorphic; i.e., one is the GIT compactifi-cation of the configurations of six lines, and the other is the Baily-Borel-Satakecompactification of the lattice polarized K3 surfaces. Due to this nice property, wecan associate geometry to each point in the moduli space M . We expect that anice geometry of degenerations, e.g. [13, 14], will arise from the boundary pointswhich we have constructed in the resolutions of M . In particular, it is an inter-esting problem to see how the geometry of the geometric mirror symmetry due toStrominger-Yau-Zaslow [27] (and also [13]) appears near these boundary points.We should note here that the standard mirror symmetry for the lattice polarizedK3 surfaces [7] does not apply to the double cover family of K3 surfaces becausethe transcendental lattice contains U (2) instead of U (cf. [15]).Finally, we note that the hypergeometric system E (3 , is a special case ofAomoto-Gel’fand systems, which are called hypergeometric system E ( n, m ) onGrassmannians G ( n, m ) (see e.g. [2] and refereces therein). Our theorems aboveare based on explicit constructions for the case of E (3 , , but we expect that theyare generalized in the following form: Conjecture 7.4. Hypergeometric D -modules of E ( n, m ) on Grassmannians havesimilar coverings by the D -modules of suitable GKZ systems. Namely, the parameterspace of the system E ( n, m ) has an open covering by Zariski open subset of toricvarieties on which the system is represented (locally) by a GKZ system. The cases of E ( n, n ) are related to Calabi-Yau varieties which are given by(suitable resolutions of) the double coverings of P n − branched along general n -hyperplanes. In particular, the case of E (4 , and its related algebraic geometryhas been worked in the literatures [12, 26]. In this case, the GIT quotient parameterspace for E (4 , and its toric covering by M SecP for the GKZ system become muchmore complicated. However, we expect similar results as in Theorems 7.1,7.2 holdin general. ppendix A. Six singular vertices of Sec ( A ) The secondary polytope Sec ( A ) is defined for the Gorenstein cone Cone ( A ) gen-erated by primitive generators A = { v , v , ..., v } given in (2.9). We first considerall possible (regular) triangulations T of the convex hull Conv ( A ) . Each triangu-lation T = { σ } consists of simplices σ , each of which corresponds to a simplicialcone in Cone ( A ) . For a triangulation T = { σ } , we set ψ T = (cid:18) X v ≺ σ vol ( σ ) , X v ≺ σ vol ( σ ) , · · · , X v ≺ σ vol ( σ ) (cid:19) ∈ Z . Here vol ( σ ) is the volume of σ normalized so that the elementary simplex in R n is . The secondary polytope is defined to be the convex hull Conv ( { ψ T } T ∈T ) in R . By translating one vertex, say ψ T , to the origin, this polytope now sits in L R as introduced in Subsection (2.4.b). There are 108 triangulations for Conv ( A ) .Of those exactly six triangulations T , T , ..., T correspond to singularities in thecompactification M SecP = P Sec ( A ) . Below we list the all six vertices ψ T i − ψ T ∈ L for the convex hull; ψ T − ψ T = 4 ( 0 , , , , , , , , ,ψ T − ψ T = 4 ( − , , − , , , , , , ,ψ T − ψ T = 4 ( − , − , , , , , , , ,ψ T − ψ T = 4 ( 0 , − , − , , , , , , ,ψ T − ψ T = 4 ( − , − , − , , , , , , ,ψ T − ψ T = 4 ( − , − , − , , , , , , . The factor 4 is irrelevant to define toric variety from the convex hull. Put V := (cid:26) 14 ( ψ T i − ψ T ) | i = 1 , ..., (cid:27) . Note that the set V \ { } represents exactly the exponents of x, y, z, u, v in (4.1).The cone generated by V is C given in (4.3), while the cone generated by all 108vertices is C NE given in Proposition 2.5, i.e., C NE = X i =1 R ≥ ( ψ T i − ψ T ) . Appendix B. Four dimensional cones C and C NE Let L = Ker (cid:8) ϕ A : Z A → Z (cid:9) be the lattice defined in (2.10). Here, for conve-nience, we summarize the data of the cones C , C NE and their duals, which arescattered in the text. We define a projection(B.1) π : L → Z , ℓ = ( ℓ , ..., ℓ , ℓ , ..., ℓ , ..., ℓ ) ( ℓ , ..., ℓ ) . It is an easy exercise to verify that π : L → Z is an isomorphism. In thispaper we shall often use π to represent vertices in L as four component vectors forcomputations. roposition B.1. The cones C ⊂ C NE and C NE in L R are written under theabove identification by C = Cone { (0 , , , , (1 , , , , (0 , , , , (1 , , , , (0 , , , } ,C NE = Cone (cid:26) (1 , , , , (0 , , − , , (1 , − , , , (0 , , , , , , − , ( − , , , (cid:27) . It is straightforward to verify the following results from explicit calculations. Proposition B.2. The dual cones C ∨ ⊃ C ∨ NE are written by the following primitivegenerators; C ∨ = Cone (cid:26) (0 , , , , (1 , , , , (0 , , , , (0 , , , , , , − , ( − , , , (cid:27) ,C ∨ NE = Cone { (1 , , , , (0 , , , , (0 , , , , (1 , , , , (0 , , , } . The dual cone C ∨ is a Gorenstein cone, while C ∨ NE is not. Appendix C. Picard-Fuchs operators on Spec C [( σ (1)2 ) ∨ ∩ L ] We list the Picard-Fuchs differential operators discussed in Subsection 3.2 fol-lowing the notation there. A complete set of differential operators D ℓ are given bythe following ℓ ’s: ℓ (1) , ℓ (2) , ℓ (3) , ℓ (1) + ℓ (4) , ℓ (2) + ℓ (4) , ℓ (3) + ℓ (4) ,ℓ (1) + ℓ (2) + ℓ (4) , ℓ (1) + ℓ (3) + ℓ (4) , ℓ (2) + ℓ (3) + ℓ (4) . We name by D i ( i = 1 , ..., the associated operators D ℓ in the above order of ℓ with setting z i := a ℓ ( i ) and θ i := z i ∂∂z i . They take the following forms: D = ( θ + θ − θ )( θ + θ − θ ) + z ( θ + )( θ − θ ) , D = ( θ + θ − θ )( θ + θ − θ ) + z ( θ + )( θ − θ ) , D = ( θ + θ − θ )( θ + θ − θ ) + z ( θ + )( θ − θ ) , D = ( θ − θ )( θ − θ ) − z z ( θ + )( θ + θ − θ ) , D = ( θ − θ )( θ − θ ) − z z ( θ + )( θ + θ − θ ) , D = ( θ − θ )( θ − θ ) − z z ( θ + )( θ + θ − θ ) , D = ( θ + θ − θ )( θ − θ ) + z z z ( θ + )( θ + ) , D = ( θ + θ − θ )( θ − θ ) + z z z ( θ + )( θ + ) , D = ( θ + θ − θ )( θ − θ ) + z z z ( θ + )( θ + ) . The radical √ dis of the discriminant is given by z z z z × Y i =1 (1 + z i )(1 + z i z ) × Y ≤ i Birational map φ : M , M Here we describe the birational map φ : M , M explicitly by coordinates.We follow the general definitions given in [6, 24]. C.1. Semi-invariants for M . As in the text, let us consider an ordered configu-ration of six lines ( ℓ ℓ ...ℓ ) by the corresponding sequence of points A = ( a a ... a ) epresented by a × matrix. Based on the classical invariant theory, following[6], we define the following homogeneous polynomials(D.1) Y = Y ( A ) = [1 2 3][4 5 6] ,Y = Y ( A ) = [1 2 4][3 5 6] ,Y = Y ( A ) = [1 2 5][3 4 6] ,Y = Y ( A ) = [1 3 4][2 5 6] ,Y = Y ( A ) = [1 3 5][2 4 6] ,Y = Y ( A ) = [1 2 3][1 4 5][2 4 6][3 5 6] − [1 2 4][1 3 5][2 3 6][4 5 6] , where [ i j k ] := det( a i a j a k ) , and we count the weight Y , ..., Y by 1 and Y by2 since they are sections of L and L ⊗ , respectively, for a GL (3 , C ) × ( C ∗ ) -equivariant line bundle L with the fiber C det ⊗ C (1 ) . The GIT quotient GL (3 , C ) \ M (3 , ss / ( C ∗ ) coincides with the Zariski closure of the image A [ Y , Y , ..., Y ] in the weighted projective space P (1 , , which we have denoted by M in the text.From symmetry reason, we extend the weight one variables Y , ..., Y to Y = [1 2 6][3 4 5] , Y = [1 3 6][2 4 5] , Y = [1 4 6][2 3 5] ,Y = [1 5 6][2 3 4] , Y = [1 4 5][2 3 6] . These satisfy the following linear relations, which are nothing but Plücker relationsof the Grassmannian G (3 , :(D.2) Y − Y + Y − Y = 0 , Y − Y + Y − Y = 0 ,Y − Y − Y + Y = 0 , Y − Y − Y + Y = 0 ,Y − Y + Y + Y = 0 . C.2. Semi-invariants for M , . When we write an ordered 6 lines in generalposition by A = ( a a ... a ) as above, the birational map (2.6) may be expressedby A A ∗ = ( a × a a × a a × a a a a ) =: ( c c c a a a ) , where a i × a j represents the exterior product of two space vectors a i , a j . Similarlyto the case of A , two algebraic groups GL (3 , C ) and ( C ∗ ) act on the column vectorsof A ∗ , but with different representations. This time, the semi-invariants are givenby(D.3) X = X ( A ∗ ) = ( c , a )( c , a )( c , a ) ,X = X ( A ∗ ) = ( c , a )( c , a )( c , a ) ,X = X ( A ∗ ) = ( c , a )( c , a )( c , a ) ,X = X ( A ∗ ) = ( c , a )( c , a )( c , a ) ,X = X ( A ∗ ) = ( c , a )( c , a )( c , a ) ,X = X ( A ∗ ) = ( c , a )( c , a )( c , a ) , with ( x , y ) := P i =1 x i y i . Using these semi-invariants, the GIT quotient of theconfiguration space of 3 points and 3 lines in P coincides with the Zariski closureof the image A ∗ [ X , X , ..., X ] in P , which is the toric variety M , . C.3. The birational map φ : M , M and S actions. The birationalmap (2.6) can be written explicitly by eliminating the variables a i from (D.1) and D.3). Using a Gröbner basis package in symbolic manipulations, we obtain(D.4) Y = X + X + X − X − X − X ,Y = X − X ,Y = X − X ,Y = X − X ,Y = X − X ,Y = X X + X X + X X − X X − X X − X X , which represents the birational map φ : M , M . The inverse rational map φ − takes the following form:(D.5) X = 12 Y ( Y ( Y − Y + Y + Y + Y ) − Y Y + Y Y + Y ) ,X = 12 Y ( Y ( Y + Y + Y + Y − Y ) − Y Y + Y Y + Y ) ,X = 12 Y ( Y ( − Y + Y − Y − Y + Y ) − Y Y + Y Y + Y ) ,X = 12 Y ( Y ( − Y + Y + Y − Y + Y ) − Y Y + Y Y + Y ) ,X = 12 Y ( Y ( − Y + Y − Y + Y + Y ) − Y Y + Y Y + Y ) ,X = 12 Y ( Y ( Y − Y + Y + Y − Y ) − Y Y + Y Y + Y ) . Appendix E. Singular lines in M Here, for convenience of the reader, we list the ideals for 15 lines in M . Sinceall lines are in the hyperplane Y = 0 , we omit Y in each ideal. h Y − Y , Y − Y , Y + Y − Y i , h Y , Y , Y − Y + Y i , h Y , Y , Y − Y + Y i , h Y , Y , Y + Y i , h Y , Y , Y + Y i , h Y , Y , Y − Y i , h Y , Y , Y − Y i , h Y , Y , Y + Y − Y i , h Y , Y , Y + Y − Y i ; h Y , Y − Y , Y − Y i , h Y , Y , Y i , h Y , Y − Y , Y − Y i , h Y , Y , Y i , h Y , Y , Y i , h Y , Y , Y i . We write these lines by L , ..., L ; L , ..., L in order. The first 9 lines correspondto the singular lines in M , under the birational map φ : M , M . As for thelast 6 lines, which lie on { Y = 0 } , we can verify that the inverse images of theselines are planes in M , which are given by P ijk = { X = X i , X = X j , X = X k } ⊂ M , for 6 permutations ( ijk ) of (3 , , . Appendix F. Properties used in the proof of Theorem 6.12 We prove the properties used in the proof of Theorem 6.12. We continue usingthe notation introduced there. laim F.1. The following assertions hold: (1) − ( K f M + 1 / e E ) is f -nef, and is numerically f -trivial only for e ℓ , . . . , e ℓ . (2) There exists a small contraction ρ : f M → M over M contracting exactly e ℓ ∪ · · · ∪ e ℓ . (3) The contraction ρ is a log flipping contraction with respect to some klt pair ( f M , D ) . (4) The flip f M f M +6 of ρ exists and it coincides locally with the flip con-structed as in Proposition 4.4.Proof. (1) Note that K M ′ = f ∗ K M + E since f is the blow-up along Sing M and M has ODP generically along Sing M . Let F be the f -exceptional divisor. Wehave K f M = f ∗ K M + F since f is the blow-up at singular points isomorphic tothe vertex of the cone over the Segre ( P ) . Therefore we have − ( K f M + 1 / e E ) = − ( f ∗ f ∗ K M + 1 / e E + f ∗ E + F ) . Now note that f ∗ E = e E + F , which follows fromthe local computations as in the proof of Proposition 5.2 (note that, in the proof ofProposition 5.2, we can read off that the divisor E is defined by u = 0 on the chartof e X with U = 1 ). Therefore we have − ( K f M + 1 / e E ) ≡ M − (4 / e E + 2 F ) . It is easy to see that − (4 / e E + 2 F ) is f -nef from the local computations for f and f .(2) By Proposition 5.3, ( f M , / e E ) is a klt pair. Since − ( K e X + 1 / e E ) is f -nef by (1), and also f -big, then − ( K f M + 1 / e E ) is f -semiample by Kawamata-Shokurov’s base point free theorem ([20]). Therefore, there exists a contraction ρ : f M → M over M defined by a sufficient multiple of − ( K f M + 1 / e E ) . Since − ( K f M + 1 / e E ) is numerically f -trivial only for l , . . . , l by (1), we see that ρ isthe desired contraction.(3) The proof given here may look technical but more or less is standard forexperts. As we see in the proof of (1) and (2), ( f M , / e E ) is a klt pair such that − ( K f M + 1 / e E ) is numerically ρ -trivial. Now let A , B be ample divisors on f M and M , respectively. Then we see that | mρ ∗ B − A | 6 = ∅ for m ≫ since ρ ∗ B is big. Let G be a member of | mρ ∗ B − A | . Then ( f M , / e E + 1 /k G ) is klt for k ≫ and − ( K f M + 1 / e E + 1 /k G ) is ρ -ample since − ( K f M + 1 / e E ) is numerically ρ -trivial and − G is ρ -ample. Setting D := 1 / e E + 1 /k G , we obtain a desired logpair.(4) The existence of the flip is a consequence of (3) and [4, Cor.1.4.1]. By the localuniquness of the flip [20, Prop.5-11-1], it coincides locally with the flip constructedas in Proposition 4.4. (cid:3) eferences [1] K. Aomoto, On the structure of integrals of power products of linear functions , Sci. Papers,Coll. Gen. Education, Univ. Tokyo, 27(1977), 49–61.[2] K. Aomoto and M. Kita, Theory of Hypergeometric Functions , Springer Monograph in Math-ematics, Springer (2011).[3] V. Batyrev and D.A. Cox, On the Hodge structure of projective hypersurfaces in toric vari-eties , Duke Math. J. 75 (1994) 293–338.[4] C. Birkar, P. Cascini, C. D. Hacon and J. McKernan, Existence of minimal models forvarieties of log general type , J. Amer. Math. Soc. 23 (2010), 405–468[5] W. Decker, G.-M. Greuel, G. Pfister and H. Schönemann: Singular 4-1-1 — A computeralgebra system for polynomial computations Mirror symmetry for lattice polarized K3 surfaces , Algebraic geometry, 4. J.Math. Sci. 81 (1996) 2599–2630.[8] I.M. Gel’fand and S.I. Gel’fand, Generalized Hypergeometric equations , Soviet Math. Dokl.33, (1986), 643–646.[9] I.M. Gel’fand and M.I. Graev, Hypergeometric functions associated with the Grassmannian G , , Soviet Math. Dokl. 35 (1987) 298–303.[10] I.M. Gel’fand, A. V. Zelevinski, and M.M. Kapranov, Equations of hypergeometric type andtoric varieties , Funktsional Anal. i. Prilozhen. 23 (1989), 12–26; English transl. FunctionalAnal. Appl. 23(1989), 94–106.[11] I.M. Gel’fand, A. V. Zelevinski, and M.M. Kapranov, Discriminants, resultants, and multi-dimensional determinant , Birkhäuser Boston 1994.[12] R. Gerkmann, M. Sheng, D. van Straten and K. Zuo, On the monodromy of the moduli spacesof Calabi-Yau threefolds coming from eight planes in P , Mathh. Ann. (2013)355: 187–214.[13] M. Gross and B. Siebert, Mirror symmetry via logarithmic degeneration data I . Journal ofDifferential Geometry, 72(2):169–338, 2006.[14] M. Gross and B. Siebert, Mirror symmetry via logarithmic degeneration data II , Journal ofAlgebraic Geometry, 19(4):679–780, 2010.[15] M. Gross, P.M.H. Wilson, Large complex structure limits of K3 surfaces , Journal of Differ-ential Geometry, 55(3):475–546, 2000.[16] S. Hosono, A. Klemm, S. Theisen and S.-T. Yau, Mirror Symmetry, Mirror Map and Appli-cations to complete Intersection Calabi-Yau Spaces , Nucl. Phys. B433(1995)501–554.[17] S. Hosono, B.H. Lian and S.-T. Yau, GKZ-Generalized hypergeometric systems in mirrorsymmetry of Calabi-Yau hypersurfaces , Commun. Math. Phys. 182 (1996) 535–577.[18] S. Hosono,B.H. Lian, H. Takagi and S.-T. Yau, K3 surfaces from configurations of six linesin P and mirror symmetry II — λ K -functions—, preprint to appear.[19] B. Hunt, The geometry of some special arithmetic quotients, Lect. Notes in Math. 1637 (1996)Springer-Verlag, Berlin Heidelberg.[20] Y.Kawamata, K.Matsuda, and K.Matsuki Introduction to the Minimal Model Problem , Al-gebraic Geometry, Sendai, 1985, T. Oda, ed. (Tokyo: Mathematical Society of Japan, 1987),283–360[21] K. Matsumoto, Theta functions on the bounded symmetric domain of type I , and theperiod map of a 4-parameter family of K3 surfaces, Math. Ann. 295 (1993) 383–409.[22] K. Matsumoto, T. Sasaki and M. Yoshida, The monodromy of the period map of a 4-parameter family of K3 surfaces and the hypergeometric function of type E (3 , , Internat.J. Math. vol.3, No.1 (1992) 1 –164.[23] J. Rambau, TOPCOM: Triangulations of Point Configurations and Oriented Matroids ,Mathematical Software - ICMS 2002 (Cohen, Arjeh M. and Gao, Xiao-Shan and Takayama,Nobuki, eds.), World Scientific (2002), pp. 330-340.[24] E. Reuvers, Moduli spaces of configurations Compactifications of the configuration space of six points ofthe projective plane and fundamental solutions of the hypergeometric system of type (3,6) ,Tohoku Math. J. (2) Volume 49, Number 3 (1997), 379-413. 26] M. Sheng and K. Zuo, Polarized variation of Hodge structures of Calabi-Yau type ans char-acteristic subvarieties over symmetric domains, Math. Ann. (2010)348:211–236.[27] A. Strominger, S.-T. Yau and E. Zaslow, Mirror symmetry is T-Duality , Nucl. Phys. B479(1996) 243–259.[28] G. van der Geer, On the Geometry of a Siegel Modular Threefold, Math. Ann. 260 (1982)317–350.[29] M. Yoshida, Hypergeometric Functions, My Love – Modular Interpretations of ConfigurationSpaces– , Springer, 1997.Shinobu HosonoDepartment of Mathematics, Gakushuin University,Mejiro, Toshima-ku, Tokyo 171-8588, Japane-mail: [email protected] H. LianDepartment of Mathematics, Brandeis University,Waltham MA 02454, U.S.A.e-mail: [email protected] TakagiDepartment of Mathematics, Gakushuin University,Mejiro, Toshima-ku, Tokyo 171-8588, Japane-mail: [email protected]. YauDepartment of Mathematics, Harvard University,Cambridge MA 02138, U.S.A.e-mail: [email protected], Springer, 1997.Shinobu HosonoDepartment of Mathematics, Gakushuin University,Mejiro, Toshima-ku, Tokyo 171-8588, Japane-mail: [email protected] H. LianDepartment of Mathematics, Brandeis University,Waltham MA 02454, U.S.A.e-mail: [email protected] TakagiDepartment of Mathematics, Gakushuin University,Mejiro, Toshima-ku, Tokyo 171-8588, Japane-mail: [email protected]. YauDepartment of Mathematics, Harvard University,Cambridge MA 02138, U.S.A.e-mail: [email protected]