Euler Characteristics of Crepant Resolutions of Weierstrass Models
aa r X i v : . [ m a t h . AG ] M a y Euler characteristics of crepant resolutions ofWeierstrass models
Mboyo Esole ♠ , Patrick Jefferson ♣ , Monica Jinwoo Kang ♣ [email protected], [email protected], patrickjeff[email protected] ♠ Department of Mathematics, Northeastern UniversityBoston, MA 02115, USA ♣ Department of Physics, Jefferson Physical LaboratoryHarvard University, Cambridge, MA 02138, U.S.A.
Abstract
Based on an identity of Jacobi, we prove a simple formula that computes the pushforward ofanalytic functions of the exceptional divisor of a blowup of a projective variety along a smoothcomplete intersection with normal crossing. We use this pushforward formula to derive generatingfunctions for Euler characteristics of crepant resolutions of singular Weierstrass models given byTate’s algorithm. Since the Euler characteristic depends only on the sequence of blowups and noton the Kodaira fiber itself, several distinct Tate models have the same Euler characteristic. In thecase of elliptic Calabi-Yau threefolds, using the Shioda–Tate–Wazir theorem, we also compute theHodge numbers. For elliptically fibered Calabi-Yau fourfolds, our results also prove a conjecture ofBlumenhagen, Grimm, Jurke, and Weigand based on F-theory/heterotic string duality. ontents G -models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 The pushforward theorem and Jacobi’s identity . . . . . . . . . . . . . . . . . . . . . . . 61.4 Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5 Organization of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 B.1 Lefschetz fixed point theorem and the Euler characteristic as an intersection number 34B.2 Poincaré-Hopf theorem and the Euler characteristic . . . . . . . . . . . . . . . . . . . . 35B.3 Hirzebruch–Riemann–Roch theorem and the Euler characteristic . . . . . . . . . . . . 35
C Basic Notions 37
C.1 Fiber types, dual graphs, Kodaira symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 37C.2 Elliptic fibrations, generic versus geometric fibers . . . . . . . . . . . . . . . . . . . . . . 39C.3 Weierstrass models and Deligne’s formulaire . . . . . . . . . . . . . . . . . . . . . . . . 40C.4 Tate’s algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
Introduction
The study of crepant resolutions of Weierstrass models, their fibral structure, and their flop transi-tions is an area of common interest to algebraic geometers, number theorists, and string theorists[33, 31, 32, 30, 39, 35, 69]. The theory of elliptic surfaces has its beginnings in the 1960s, andwas advanced largely by the contributions of mathematicians such as Kodaira [46]; Néron [62];Mumford and Suominen [59], Deligne [16], and Tate [70]. Miranda studied the desingularization ofelliptic threefolds and the phenomenon of collisions of singularities in [56], and Szydlo subsequentlygeneralized Miranda’s work to elliptic n -folds [69]; the Picard number (i.e., the rank of the Néron-Severi group) of an elliptic fibration can be obtained using the Shioda–Tate–Wazir theorem [73];the study of elliptic fibrations having the same Jacobian was developed by Dolgachev and Gross[21]; and Nakayama studied local and global properties of Weierstrass models over bases of arbi-trary dimension in [61, 60]. Furthermore, more recent developments have been inspired by stringtheory (in particular, M-theory and F-theory) constructions that ascribe an interesting physicalmeaning to various topological and geometric properties of elliptically-fibered Calabi-Yau varieties[71, 57, 58, 8, 42, 17].A Weierstrass model provides a convenient framework for computing the discriminant, the j -invariant, and the Mordell–Weil group of an elliptic fibration. Weierstrass models are also the settingin which Tate’s algorithm is defined [70]. Any elliptic fibration over a smooth base is birational toa (potentially singular) Weierstrass model [16]. Since a Weierstrass model is a hypersurface, it isGorenstein [22, Corollary 21.19], and hence its canonical class is well-defined as a Cartier divisor.In practice, it is often necessary to regularize the singularities of Weierstrass models when com-puting, for example, their topological invariants. Among the possible regularizations of a singularvariety, crepant resolutions are particularly desirable as, by definition, they preserve the canonicalclass and the smooth locus of the variety. In a sense, crepant resolutions modify the variety as mildlyas possible while regularizing its singularities. Surfaces with canonical singularities always have acrepant resolution, which is unique up to isomorphism. However, for varieties of dimension threeor higher, crepant resolutions do not necessarily exist, and when they do, they may not be unique.Distinct crepant resolutions of the same Weierstrass model are connected by a network of flops. Example 1.1.
The quadric cone over a conic surface V ( x x − x x ) ⊂ C has two crepant resolutionsrelated by an Atiyah flop. By contrast, the quadric cone V ( x + x + x + x + x ) ⊂ C doesnot have a crepant resolution since it has Q -factorial terminal singularities. The binomial variety V ( x x − u u u ) ⊂ C has six crepant resolutions whose network of flops forms a hexagon [33]. Foradditional examples of flops involving Weierstrass models, see [31, 32, 27, 28].There is an important subset of singular Weierstrass models that have crepant resolutions andplay a central role in string geometry, as they are instrumental in the geometric engineering of gaugetheories in F-theory and M-theory. We refer to them as G -models, they are defined in §1.2 and aretypically obtained by the Weierstrass models that appear as outputs of Tate’s algorithm [70, 8, 45].The networks of crepant resolutions of these Weierstrass models are conjectured to be isomorphicto the incidence graph of the chambers of a hyperplane arrangement [42, 39, 25, 26].The number of distinct resolutions associated to a G -model can be rather large [25, 26, 39]. It isinteresting to study topological invariants that do not depend on the choice of a crepant resolution.2n example of such a topological invariant is the Euler characteristic—using p -adic integration andWeil conjecture, Batyrev proved that the Betti numbers of smooth varieties connected by a crepantbirational map are the same [6], and it therefore follows that the Euler characteristics of any twocrepant resolutions are the same.The purpose of this paper is to compute the Euler characteristics of G -models obtained bycrepant resolutions of Weierstrass models, where G is a simple group. Following [2, 3], we allow thebase to be of arbitrary dimension and we do not impose the Calabi-Yau condition. We work relativeto a base that we leave arbitrary. In this sense, our paper is a direct generalization of the work ofFullwood and van Hoeij on stringy invariants of Weierstrass models [35].The Euler characteristic of an elliptic fibration plays a central role in many physical problems suchas the computation of gravitational anomalies of six dimensional supergravity theories [37, 63] andthe cancellation of tadpoles in four dimensional theories [66, 2, 3, 9, 14, 24, 29]. Unfortunately, theEuler characteristics of crepant resolutions of Weierstrass models are generally not known, althoughthey have been computed in some special cases for Calabi-Yau threefolds and fourfolds [4, 5, 53, 35].For instance, the Euler characteristics of G -models for Calabi-Yau threefolds were studied in [37],and there are conjectures for the Euler characteristics of G -models for Calabi-Yau fourfolds basedon heterotic string theory/F-theory duality [9].As a byproduct of our results, we prove a conjecture by Blumenhagen, Grimm, Jurke, andWeigand [9] on the Euler characteristics of Calabi-Yau fourfolds which are G -models for G = SU ( ) ,SU( ), SU( ), SU( ), E , E or E . These groups correspond to the exceptional series E k defined onpage 9 with the exception of D . In [9], the authors conjecture the value of the Euler characteristicusing a method inspired by heterotic string theory/F-theory duality. The results of our computationmatch their prediction precisely, except for the limiting case of the group E . We also retrieve knownresults for the case of G -models that are Calabi-Yau threefolds [37], while removing most of theassumptions of [37].A crucial ingredient of our results is Theorem 1.8, which is a pushforward formula for any analyticfunction of the class of the exceptional divisor of a blowup of a nonsingular variety along a smoothcomplete intersection of hypersurfaces meeting transversally. Theorem 1.8 is a generalization toarbitrary analytic functions of a result of Fullwood and van Hoeij [35, Lemma 2.2 ], which relies ona theorem of Aluffi [1] simplifying the classic formula of Porteous on Chern classes of the tangentbundle of a blowup [64]. Theorem 1.8 profoundly simplifies the algebraic manipulations necessary tocompute pushforwards, and therefore has a large range of applications independently of the specificapplications discussed in this paper.For the reader’s convenience, we provide tables specializing our results to the cases of ellipticthreefolds and fourfolds, and further to the cases of Calabi-Yau threefolds and fourfolds, includingan explicit computation of the Hodge numbers in the Calabi-Yau threefold case. We emphasize thatour results are insensitive to the particular choice of a crepant resolution due to Batyrev’s theoremon the Betti numbers of crepant birational equivalent varieties [6] and Kontsevich’s theorem on theHodge numbers of birational equivalent Calabi-Yau varieties [47].3 .1 Conventions Throughout this paper, we work over the field of complex numbers. A variety is a reduced and irre-ducible algebraic scheme. We denote the vanishing locus of the sections f , . . . , f n by V ( f , . . . , f n ) .The tangent bundle of a variety X is denoted by T X and the normal bundle of a subvariety Z of avariety X is denoted by N Z X . Let V → B be a vector bundle over a variety B . We denote the by P ( V ) the projective bundle of lines in V . We use Weierstrass models defined with respect to theprojective bundle π ∶ X = P [ O B ⊕ L ⊗ ⊕ L ⊗ ] → B where L is a line bundle of B . We denotethe pullback of L with respect to π by π ∗ L . We denote by O X ( ) the canonical line bundle on X , i.e., the dual of the tautological line bundle of X (see [36, Appendix B.5]). The first Chernclass of O X ( ) is denoted H and the first Chern class of L is denoted L . The Weierstrass model ϕ ∶ Y → B is defined as the zero-scheme of a section of O X ( ) ⊗ π ∗ L ⊗ —Weierstrass models arestudied in more detail in §C.3. The Chow group A ∗ ( X ) of a nonsingular variety X is the groupof divisors modulo rational equivalence [36, Chap. 1,§1.3]. We use [ V ] to refer to the class of asubvariety V in A ∗ ( X ) . Given a class α ∈ A ∗ ( X ) , the degree of α is denoted ∫ X α (or simply ∫ α if X is clear from the context.) Only the zero component of α is relevant in computing ∫ X α —see [36,Definition 1.4, p. 13]. We use c ( X ) = c ( T X ) ∩ [ X ] to refer to the total homological Chern class ofa nonsingular variety X , and likewise we use c i ( T X ) to denote the i th Chern class of the tangentbundle T X . Given two varieties
X, Y and a proper morphism f ∶ X → Y , the proper pushforwardassociated to f is denoted f ∗ . If g ∶ X → Y is a flat morphism, the pullback of g is denoted g ∗ andby definition g ∗ [ V ] = [ g − ( V )] , see [36, Chap 1, §1.7]. Given a formal series Q ( t ) = ∑ ∞ i = Q i t i , wedefine [ t n ] Q ( t ) = Q n .Our conventions for affine Dynkin diagrams are as follows. A projective Dynkin diagram isdenoted M k where M is A , B , C , D , E , F , or G , and k is the number of nodes. An affine Dynkindiagram that becomes a projective Dynkin diagram g after removing a node of multiplicity one isdenoted ̃ g . We denote by ̃ g t the (the possibly twisted) affine Dynkin diagram whose Cartan matrixis the transpose of the Cartan matrix of ̃ g . The graph of ̃ g t is obtained by exchanging the directionsof all the arrows of ̃ g . When the extra node is removed, the dual graph of ̃ g t reduces to the dualgraph of the Langlands dual of g . The affine Dynkin diagrams ̃ g t and ̃ g are distinct only when g isnot simply laced (i.e., when g is G , F , B k , or C k ). The notation ̃ g t follows Carter [13, Appendix,p. 540-609] and is equivalent to the notation ̃ g ∨ used by MacDonald in §5 of [52]. The multiplicitiesdefine a zero vector of the extended Cartan matrix. In the notation of Kac [44], ̃ B tℓ ( ℓ ≥ ), ̃ C tℓ ( ℓ ≥ ), ̃ G t , and ̃ F t are respectively denoted A ( ) ℓ − , D ( ) ℓ + , D ( ) , and E ( ) ; while ̃ B ℓ ( ℓ ≥ ), ̃ C ℓ ( ℓ ≥ ), ̃ G , and ̃ F are respectively denoted B ( ) ℓ , C ( ) ℓ , G ( ) , and F ( ) . When g is non-simply laced, theaffine Dynkin diagrams ̃ g t and ̃ g differ from each by the directions of their arrows and also by themultiplicities of their nodes (see Figure 1.1).Given a complete intersection Z of hypersurfaces Z i = V ( z i ) in a variety X , we denote the blowup ̃ X = Bl Z X of X along Z with exceptional divisor E = V ( e ) as X ̃ X. ( z , . . . , z n ∣ e ) There is a typo on page 570 of [13] in the first Dynkin diagram of ̃ B ℓ on the top of the page, where the arrow isin the wrong direction but correctly oriented in the rest of the page. B t + ℓ
11 2 2 2 1 11 2 2 2 2 ̃ B + ℓ ̃ C t + ℓ ̃ C + ℓ ̃ F t ̃ F ̃ G t ̃ G Figure 1.1: Twisted affine Lie algebras vs affine Lie algebras for non-simply laced cases. Only thoseon the left appears in the theory of elliptic fibrations as dual graphs of the fiber over the genericpoint of an irreducible component of the discriminant locus. G -models In this section, we recall how a Lie group is naturally associated with an elliptic fibration andintroduce the notion of a G -model. Our notation for dual graphs and Kodaira fibers is spelled outin §1.1, and Tables 2 and 3. See also Appendix C for the definitions of a fiber type , a generic fiber ,and a geometric generic fiber . Definition 1.2 ( K -model) . Let K be the type of a generic fiber. Let S ⊂ B be a smooth divisor ofa projective variety B . An elliptic fibration ϕ ∶ Y Ð→ B over B is said to be a K -model if1. The discriminant locus ∆ ( ϕ ) contains as an irreducible component the divisor S ⊂ B .2. The generic fiber over S is of type K .3. Any other fiber away from S is irreducible.If the dual graph of K corresponds to an affine Dynkin diagram of type ̃ g t , where g is a Lie algebra,then the K -model is also called a g -model .In F-theory, a Lie group G ( ϕ ) attached to a given elliptic fibration ϕ ∶ Y Ð→ B depends on thetype of generic singular fibers and the Mordell-Weil group MW ( ϕ ) of the elliptic fibration [10]. TheLie algebra g associated to the elliptic fibration is then the Langlands dual g ∨ = ⊕ i g ∨ i of g = ⊕ i g i .If we denote by exp ( g ∨ ) the unique (up to isomorphism) simply connected compact simple groupwhose Lie algebra is g ∨ , then the group associated to the elliptic fibration ϕ ∶ Y Ð→ B is: G ( ϕ ) ∶ = exp ( g ∨ ) MW tor ( ϕ ) × U ( ) rk MW ( ϕ ) , rk MW ( ϕ ) is the rank of the Mordell-Weil group of ϕ and MW tor ( ϕ ) is the torsion subgroupof the Mordell-Weil group of ϕ . Defining properly the quotient of exp ( g ∨ ) by the Mordell–Weilgroup requires a choice of embedding of the Mordell–Weil group in the center of exp ( g ∨ ) [55]. Definition 1.3 ( G -model) . An elliptic fibration ϕ ∶ Y Ð→ B with an associated Lie group G = G ( ϕ ) is called a G -model .If the reduced discriminant locus has a unique irreducible component S over which the genericfiber is not irreducible, the group G ( ϕ ) is simple. The relevant fiber ̃ g t can be realized by resolvingthe singularities of a Weierstrass model derived from Tate’s algorithm. The relation between thefiber type and the group G ( ϕ ) is not one-to-one. For example, an SU(2)-model can be given by adivisor S with a fiber of type I s , I ns , III, IV ns , or I ns . For that reason, a given decorated Kodairafiber provides a more refined characterization of a G -model. Example 1.4.
For n ≥ , an SU( n )-model is a I s n -model with a trivial Mordell-Weil group. For n ≥ ,a Spin(8+2 n )-model is an I ∗ s n -model with trivial Mordell-Weil group. For n ≥ , a Spin(7+2 n )-modelis an I ∗ ns n -model with trivial Mordell-Weil group. A G -model is an I ∗ ns -model with a trivial Mordell-Weil group. A Spin(7)-model is an I ∗ ss -model with a trivial Mordell-Weil group. Example 1.5 (See [28]) . The SO( ), SO( ), SO( ), and SO( )-models are respectively I ns , I ns , I s ,and I *ss -models with MW= Z / Z . For n ≥ , an SO( + n )-model is an I *s n -model with a Mordell-Weil group MW= Z / Z . For n ≥ , an SO( + n )-model is an I *ns n -model with Mordell-Weil groupMW= Z / Z . Example 1.6.
If the Mordell-Weil group is trivial, K -models with K = I s , I ns , III, IV ns , or I ns , areall SU(2)-models. An A -model can be given by a IV s -model or a I -model. If the Mordell-Weilgroup is trivial, both a IV s -model or a I s -model give a SU(3)-model. A C ℓ -model can be givenby an I ns ℓ + -model or an I ns ℓ + -model, and if the Mordell-Weil group is trivial, these both give aUSp( ℓ )-model. Remark 1.7.
Not all singular Weierstrass models are G -models as the reducible singular fibersmight not appear in codimension one. See, for example, the Jacobians of the elliptic fibrationsdiscussed in [2, 3, 24, 29]. As explained earlier, one of our key results is a pushforward theorem that streamlines all the com-putations of this paper. We present the pushforward theorem in this subsection.
Theorem 1.8.
Let the nonsingular variety Z ⊂ X be a complete intersection of d nonsingularhypersurfaces Z , . . . , Z d meeting transversally in X . Let E be the class of the exceptional divisorof the blowup f ∶ ̃ X Ð→ X centered at Z . Let ̃ Q ( t ) = ∑ a f ∗ Q a t a be a formal power series with Q a ∈ A ∗ ( X ) . We define the associated formal power series Q ( t ) = ∑ a Q a t a whose coefficients pullbackto the coefficients of ̃ Q ( t ) . Then the pushforward f ∗ ̃ Q ( E ) is: f ∗ ̃ Q ( E ) = d ∑ ℓ = Q ( Z ℓ ) M ℓ , where M ℓ = d ∏ m = m ≠ ℓ Z m Z m − Z ℓ .
6e call the coefficient M ℓ the ℓ -moment of the blowup f . Remark 1.9.
Given a blowup f ∶ ̃ X Ð→ X , any element α of the Chow ring A ∗ ( ̃ X ) can be expressedas α = ∑ ∞ n = f ∗ α i E i where α i are elements of the Chow ring A ∗ ( X ) . So Theorem 1.8 can be used topushforward any element of A ∗ ( ̃ X ) .Theorem 1.8 is proven in §3. By the projection formula and the linearity of the pushforward, theproof of Theorem 1.8 is almost trivial once it is established in the special case of a monic monomial Q ( t ) = t k . This special case is Lemma 3.7 on page 18. The proof of the Lemma 3.7 relies on anidentity due to Carl Gustave Jacobi that gives a partial fraction formula for homogeneous completesymmetric polynomials: Lemma 1.10 (Jacobi) . Let h r ( x , . . . , x d ) be the homogeneous complete symmetric polynomial ofdegree r in d variables of an integral domain. Then: h r ( x , ⋯ , x d ) = d ∑ ℓ = x r + d − ℓ d ∏ m = m ≠ ℓ x ℓ − x m . Jacobi first proved this identity in 1825 in a slightly different form in his doctoral thesis asa partial fraction reformulation of the generating function of complete homogeneous polynomials.Lemma 1.10 was rediscovered in many different mathematical and physical problems, as discussedelegantly in [38]. For example, a proof using Schur polynomials was proposed as the solution toExercise 7.4 of [68]. For a proof using integrals and residues see Appendix A of [51]; for a proofusing matrices, see [15]. We give a short and simple proof of this identity in Appendix A.We also make use of a second pushforward theorem that concerns the projection from the ambientprojective bundle to the base B over which the Weierstrass model is defined. Let V be a vectorbundle of rank r over a nonsingular variety B . The Chow ring of a projective bundle π ∶ P ( V ) Ð→ B is isomorphic to the module A ∗ ( B )[ ζ ] modded out by the relation [36, Remark 3.2.4, p. 55] ζ r + c ( π ∗ V ) ζ r − + ⋯ + c i ( π ∗ V ) ζ r − i + ⋯ + c r ( π ∗ V ) = , ζ = c ( O P ( V ) ( )) . Theorem 1.11 (See [2, 3, 34]) . Let L be a line bundle over a variety B and π ∶ X = P [ O B ⊕ L ⊗ ⊕ L ⊗ ] Ð→ B a projective bundle over B . Let ̃ Q ( t ) = ∑ a π ∗ Q a t a be a formal power series in t suchthat Q a ∈ A ∗ ( B ) . Define the auxiliary power series Q ( t ) = ∑ a Q a t a . Then π ∗ ̃ Q ( H ) = − Q ( H ) H ∣ H =− L + Q ( H ) H ∣ H =− L + Q ( ) L , where L = c ( L ) and H = c ( O X ( )) is the first Chern class of the dual of the tautological linebundle of X . [43, Section III.17, p. 29-30], Jacobi asserts: ∏ i x − a i = ∑ i x − a i ∏ ℓ ≠ i a ℓ − a i roof. Using the functoriality of Segre classses, we can write π ∗ ( − H ) = ( + L )( + L ) = − + L + + L , which can be expanded on the both sides. This gives the following expressions for the pushforwardof each power of H : π ∗ = , π ∗ H = , π ∗ H i + = [− (− ) i + (− ) i ] L i where i is nonnegative. Then, expanding Q ( H ) as a power series with coefficients in A ∗ ( B ) , ̃ Q ( H ) = ∞ ∑ i = π ∗ α i H i = π ∗ α + ( π ∗ α ) H + H ∞ ∑ k = ( π ∗ α k ) H k , the pushforward of Q ( H ) can hence be computed as π ∗ ̃ Q ( H ) = − ∞ ∑ k = α k (− L ) k + ∞ ∑ k = α k (− L ) k = − Q ( H ) − α H − α H ∣ H =− L + Q ( H ) − α H − α H ∣ H =− L = − Q ( H ) H ∣ H =− L + Q ( H ) H ∣ H =− L + Q ( ) L . We take an intersection theory point of view inspired by Fulton [36] and Aluffi [1], and use explicitcrepant resolutions of Tate models to compute their Euler characteristics. Using Chern classes,we evaluate the Euler characteristic without dealing with the combinatorics of the fiber structure.Instead, we compute the pushforward of the homological Chern class of the variety to the base of thefibration. Since the Euler characteristics of two crepant resolutions of the same Weierstrass modelare the same [6], we do not need to explore the network of all flops to arrive at our conclusions.Our method for computing the Euler characteristics of G -models is as follows. Given a choice ofLie group G , we first use Tate’s algorithm to determine a singular Weierstrass model Y Ð→ B suchthat G is the Lie group attached to the elliptic fibration following the F-theory algorithm discussedin §1.2. We then determine a crepant resolution f ∶ Y Ð→ Y of the singular Weierstrass modelto obtain an explicit realization of the G -model as a smooth projective variety. By doing so, weretrieve the data necessary to compute the total homological Chern class of the crepant resolution f ∶ Y Ð→ Y . We apply Theorem 1.8 repeatedly to push this class forward to the projective bundle X in which the Weierstrass model is defined. Finally, we use Theorem 1.11 to push the total Chernclass forward to B . In doing so, we obtain a generating function of the form χ ( Y ) = ∫ B Q ( L, S ) c ( B ) , c ( B ) ∶ = c ( T B ) ∩ [ B ] , A A D A A G B D F E E E I s , I ns , III, IV ns , I ns IV s , I s I s I ns I s I ∗ ns I ∗ ss I ∗ s IV ∗ ns I ∗ s IV ∗ s III ∗ II ∗ A A A C A G B D F D E E E Table 1: Models studied in this paper.where ∫ B indicates the degree, Q ( L, S ) is a rational function in L and S such that Q ( L, ) = L + L c ( B ) .Q ( L, ) is the generating function for the Euler characteristic of a smooth Weierstrass model [2]. Therational expression Q ( L, S ) c ( B ) is defined in the Chow ring A ∗ ( B ) of the base. The expression χ ( Y ) is a generating function in the following sense. If the base has dimension d , the Euler characteristicis then given by the coefficient of t d in a power series expansion in the parameter t : χ ( Y ) = [ t d ] ( Q ( tL, tS ) c t ( T B )) , where d ∶ = dim B, where [ t n ] g ( t ) = g n for a formal series g ( t ) = ∑ ∞ i = g i t i , and c t ( T B ) = + c ( T B ) t + ⋯ + c d ( T B ) t d , is the Chern polynomial of the tangent bundle of B .It follows from the adjunction formula that one can further impose the Calabi-Yau condition bysetting L = c ( T B ) ; see Tables 8 and 9 for the Euler characteristics of elliptic threefold and fourfold G -models.In Table 1, we organize the Lie algebras associated to our choices of Tate models into a network,where an arrow indicates inclusion as a subalgebra. As is evident from Table 1, the results of thispaper cover all instances of Kodaira fibers with the exception of the general cases of I k and I ∗ k thatwill be discussed in a follow up paper. In particular, our list contains:• G -models corresponding to Deligne exceptional series: { e } ⊂ A ⊂ A ⊂ G ⊂ D ⊂ F ⊂ E ⊂ E ⊂ E . • G -models for the extended exceptional series : { e } ⊂ A ⊂ A ⊂ A ⊂ E ⊂ E ⊂ E ⊂ E ⊂ E . We recall that the Dynkin diagram of E n is the same as A n but with the n th node connected with the thirdnode. In particular, E ≅ A , E ≅ D , E = A × A , E = A , and E = A . G -models for simple orthogonal groups of small rank : { e } ⊂ SO ( ) ⊂ SO ( ) ⊂ SO ( ) . • G -models of the I ∗ series [27]: { e } ⊂ G ⊂ Spin ( ) ⊂ Spin ( ) . The remainder of the paper is organized as follows. In Section 2 we discuss some general propertiesof the Euler characteristic of an elliptic fibration. In Section 3 we discuss the pushforward theoremand explain the details of our computation of the Euler characteristic. Section 4 then describes howthese results can be used to calculate the Hodge numbers of Calabi-Yau threefold G -models. InSection 5, we describe the simplest model, the SU(2)-model, as an example of our computation. Wepresent the results of our computation in a series of tables in Section 6. Finally, in Section 7 weconclude with a discussion of the computation and comment on possible future research directions.A proof of Jacobi’s partial fraction identity is given in Appendix A, an explanation of the Eulercharacteristic as the degree of the top Chern class is given in Appendix B, and some basic factsabout Kodaira fibers, elliptic fibrations, Weierstrass models and Tate’s algorithm are collected inAppendix C. These models require a Mordell-Weil group Z / Z ; see [28]. ̃ A I , II ̃ A I s , I ns , I ns , III , IV ns ̃ A ℓ − ( ℓ ≥ )
11 1 1 1 I s ℓ ̃ D + ℓ ( ℓ ≥ )
11 2 2 2 11 I ∗ s ℓ ̃ E IV ∗ s ̃ E III ∗ ̃ E II ∗ ̃ B t + ℓ ( ℓ ≥ )
11 2 2 2 1 ⎧⎪⎪⎨⎪⎪⎩ I ∗ ss for ℓ = I ∗ ns ℓ for ℓ ≥ ̃ C t + ℓ ( ℓ ≥ ) I ns + ℓ , I ns + ℓ ̃ F t IV ∗ ns ̃ G t I ∗ ns Table 2: Affine Dynkin diagrams appearing as dual graphs of decorated Kodaira fibers.11 iber Type Dual graph Dual graph of Geometric fiber ̃ A ∗ ns ℓ − ̃ B tℓ ( ℓ ≥ )
11 2 2 2 1 11 2 2 2 11I ns ℓ + ̃ C tℓ + ( ℓ ≥ ) I ns ℓ + ̃ C tℓ + ( ℓ ≥ ) IV ∗ ns ̃ F t ∗ ss ̃ B t
21 111 I ∗ ns ̃ G t
21 111
Table 3: Dual graphs for elliptic fibrations .12 ype v ( c ) v ( c ) v ( ∆ ) j Monodromy
Fiber DualGraphI ≥ ≥ C I Smooth -I ∞ ( ) ̃ A II ≥ ( − ) ̃ A III ≥ ( − ) ̃ A IV ≥ ( − − ) ̃ A I n n > ∞ ( n )
11 1 1 1 ̃ A n − I ∗ n ≥ n + ∞ (− − n − )
11 2 2 2 11 ̃ D n + ≥ n + IV ∗ ≥ (− −
11 0 ) ̃ E III ∗ ≥ ( −
11 0 ) ̃ E II ∗ ≥ ( −
11 1 ) ̃ E Table 4: Kodaira-Néron classification of geometric fibers over codimension one points of the baseof an elliptic fibration [46, 62]. The j -invariant of the I ∗ is never ∞ and can take any finite value.13 Euler Characteristic of Elliptic Fibrations
The Euler characteristic of a smooth Weierstrass model ϕ ∶ Y Ð→ B over a base B is given by thefollowing formula[2, 3] χ ( Y ) = ∫ L + L c ( B ) , where c ( B ) = c ( T B ) ∩ [ B ] is the total homological Chern class and L = c ( L ) is the first Chernclass of the fundamental line bundle L = ( R ϕ ∗ O Y ) − of the elliptic fibration. This expression isthe generating function for the Euler characteristic. Assigning weight n to the n th Chern class, theEuler characteristic of Y is the component of weight d = dim B . A direct expansion gives χ ( Y ) = − d ∑ i = ( − L ) i c d − i ( T B ) ∩ [ B ] . The Euler characteristic of an elliptic surface is given by Kodaira’s formula [46, III, Theorem 12.2,p. 14]: χ ( Y ) = ∑ i v ( ∆ i ) , where the discriminant ∆ = ∑ i ∆ i is a sum of points ∆ i and v ( ∆ i ) denotes the valuation of ∆ i . Inparticular, the Euler characteristic of the resolution of a Weierstrass model over a curve is always ∫ L : χ ( Y ) = ∫ L. There are several different ways to compute the Euler characteristic of an elliptic fibration. TheEuler characteristic (with compact support) is multiplicative on local trivial fibrations and satisfiesthe excision property ( χ ( X / Z ) = χ ( X ) − χ ( Z ) for any closed Z ⊂ X ); moreover, if φ ∶ M → N is asmooth proper morphism, then χ ( M ) = χ ( N ) χ ( N η ) where χ ( N η ) is the Euler characteristic of thegeneric fiber. It follows from these properties that the Euler characteristic of an elliptic fibrationsgets all its contribution from its discriminant locus since the Euler characteristic of a smooth ellipticcurve is zero. One can identify a partition of the discriminant locus by subvarieties V i over whichthe generic fiber is constant. The Euler characteristic is then χ ( Y ) = ∑ i χ ( V i ) χ ( Y η i ) , where Y η i is the fiber over the generic point η i of V i . This method increases quickly in complexitywhen the fiber structure becomes more involved [37].A more effective way to compute the Euler characteristic is to use the Poincaré–Hopf theorem,which asserts that the Euler characteristic of X equals the degree of the top Chern class of thetangent bundle T X evaluated on the homological class of the variety. In other words, the Eulercharacteristic is the degree of the total homological Chern class: χ ( X ) = ∫ c ( X ) , c ( X ) ∶= c ( T X ) ∩ ( X ) . This method is explained in Section 2.2 and can also be thought of as an algebraic version of the14hern–Gauss–Bonnet theorem. We give three different proofs in Appendix B.
Let X be a projective variety with at worst canonical Gorenstein singularities. We denote thecanonical class by K X . Definition 2.1.
A birational projective morphism ρ ∶ Y Ð→ X is called a crepant desingularization of X if Y is smooth and K Y = ρ ∗ K X . Definition 2.2.
A resolution of singularities of a variety Y is a proper surjective birational morphism ϕ ∶ ̃ Y Ð→ Y such that ̃ Y is nonsingular and ϕ is an isomorphism away from the singular locus of Y . In other words, ̃ Y is nonsingular and if U is the singular locus of Y , ϕ maps ϕ − ( Y ∖ U ) isomorphically onto Y ∖ U . A crepant resolution of singularities is a resolution of singularities suchthat K Y = f ∗ K X . Remark 2.3.
In dimension two, there is one and only one crepant resolution of a variety withcanonical singularities. In dimension three, crepant resolutions of Gorenstein singularities alwaysexist but are usually not unique. In dimension four or greater, crepant resolutions are not alwayspossible. However, one can always find a crepant birational morphism from a Q -factorial varietywith terminal singularities. Definition 2.4 ( D -flop (See [54, p. 156-157])) . Let f ∶ X Ð→ X a small contraction. Let D be a Q -Cartier divisor in X . A D -flop is a birational morphism f ∶ X − − → X fitting into a triangulardiagram where f and f are birational morphisms X X X f f f such that1. X i are normal varieties with at worst terminal singularities.2. f i are small contractions (i.e. their exceptional loci are in codimension two or higher).3. K X i is numerically trivial along the fibers of f i (i.e. K X i ⋅ ℓ = for any curve ℓ contracted by f i ).4. The Q -divisor − D is f -ample.5. The strict f -transform D + of D is f -ample. Definition 2.5 (flop) . The morphism f ∶ X Ð→ X is said to be a flop of f ∶ X Ð→ X if thereexists a divisor D ⊂ X such that f is a D -flop of f .15 .2 Batyrev’s theorem and the Chern class of a crepant resolution We denote the Chow ring of a nonsingular variety X by A ∗ ( X ) . The free group of generated bysubvarieties of dimension r modulo rational equivalence is denoted by A r ( X ) . The degree of a class α of A ∗ ( X ) is denoted by ∫ X α (or simply ∫ α if there is no ambiguity in the choice of X ), and isdefined to be the degree of its component in A ( X ) . The total homological Chern class c ( X ) of anynonsingular variety X of dimension d is defined by: c ( X ) = c ( T X ) ∩ [ X ] , where T X is the tangent bundle of X and [ X ] is the class of X in the Chow ring. The degree of c ( X ) is the topological Euler characteristic of X : χ ( X ) = ∫ X c ( X ) . Motivated by string geometry, Batyrev and Dais proposed in [7, Conjecture 1.3] the following con-jecture.
Conjecture 2.6 (Batyrev and Dais, see [7]) . Hodge numbers of smooth crepant resolutions of analgebraic variety defined over the complex numbers with at worse Gorenstein canonical singularitiesdo not depend on the choice of such a resolution.
Using p -adic integration and the Weil conjecture, Batyrev proved the following slightly weakerproposition: Theorem 2.7 (Batyrev, [6]) . Let X and Y be irreducible birational smooth n -dimensional projectivealgebraic varieties over C . Assume that there exists a birational rational map ϕ ∶ X − − → Y whichdoes not change the canonical class. Then X and Y have the same Betti numbers. Batyrev’s result was strongly inspired by string dualities, in particular by the work of Dixon,Harvey, Vafa, and Witten [18]. Kontsevitch proved the Batyrev–Dais conjecture for the special caseof Calabi-Yau varieties as a corollary of his newly invented theory of motivic integration; the proofrelies on Hodge theory and geometrizes Batyrev’s use of p -adic integration. Theorem 2.8 (Kontsevitch, [47]) . Let X and Y be birationally-equivalent smooth Calabi-Yau vari-eties. Then X and Y have the same Hodge numbers. As a direct consequence of Batyrev’s theorem, the Euler characteristic of a crepant resolutionof a variety with Gorenstein canonical singularities is independent on the choice of resolution. Weidentify the Euler characteristic as the degree (see Definition C.2) of the total (homological) Chernclass of a crepant resolution f ∶ ̃ Y Ð→ Y of a Weierstrass model Y Ð→ B : χ ( ̃ Y ) = ∫ c ( ̃ Y ) . We then use the birational invariance of the degree under the pushfoward to express the Eulercharacteristic as a class in the Chow ring of the projective bundle X . We subsequently push this16lass forward to the base to obtain a rational function depending upon only the total Chern class ofthe base c ( B ) , the first Chern class c ( L ) , and the class S of the divisor in B : χ ( ̃ Y ) = ∫ B π ∗ f ∗ c ( ̃ Y ) . In view of Theorem 2.7, this Euler characteristic is independent of the choice of a crepant resolution.We discuss pushforwards and their role in the computation of the Euler characteristic in more detailin Section 3.
Definition 3.1 (Pushforward, [36, Chap. 1, p. 11]) . Let f ∶ X Ð→ Y be a proper morphism. Let V be a subvariety of X , the image W = f ( V ) a subvariety of Y , and the function field R ( V ) anextension of the function field R ( W ) . The pushforward f ∗ ∶ A ∗ ( X ) → A ∗ ( Y ) is defined as follows f ∗ [ V ] = ⎧⎪⎪⎨⎪⎪⎩ if dim V ≠ dim W, [ R ( V ) ∶ R ( W )] [ V ] if dim V = dim W, where [ R ( V ) ∶ R ( W )] is the degree of the field extension R ( V )/ R ( W ) . Lemma 3.2 ([36, Chap. 1, p. 13]) . Let f ∶ X Ð→ Y be a proper map between varieties. For anyclass α in the Chow ring A ∗ ( X ) of X : ∫ X α = ∫ Y f ∗ α. Lemma 3.2 means that an intersection number in X can be computed in Y through a pushfor-ward. This simple fact has far-reaching consequences and characterizes the point of view taken inthis paper, as it allows us to express the topological invariants of an elliptic fibration in terms ofthose of the base. A formula for the Chern classes of blowups of a smooth variety along a smooth center was conjecturedby Todd and Segre and proven in the general case by Porteous [64] using the Riemann-Roch theorem.A proof using Riemann-Roch “without denominators” is presented in §15.4 of [36]. A proof withoutRiemann-Roch was derived by Lascu and Scott [48, 49]. A generalization of the formula to potentiallysingular varieties was obtained by Aluffi [1].The blowup formula simplifies dramatically when the center of the blowup is a nonsingularcomplete intersection of nonsingular hypersurfaces meeting transversally. Aluffi gives an elegantshort proof using functorial properties of Chern classes and Chern classes of bundles of tangentfields with logarithmic zeros:
Theorem 3.3 (Aluffi, [1, Lemma 1.3]) . Let Z ⊂ X be the complete intersection of d nonsingularhypersurfaces Z , . . . , Z d meeting transversally in X . Let f ∶ ̃ X Ð→ X be the blowup of X centered t Z . We denote the exceptional divisor of f by E . The total Chern class of ̃ X is then: c ( T ̃ X ) = ( + E ) ( d ∏ i = + f ∗ Z i − E + f ∗ Z i ) f ∗ c ( T X ) . (3.1) Lemma 3.4.
Let f ∶ ̃ X Ð→ X be the blowup of X centered at Z . We denote the exceptional divisorof f by E . Then f ∗ E n = ( − ) d + h n − d ( Z , ⋯ , Z d ) Z ⋯ Z d , where h i ( x , ⋯ , x k ) is the complete homogeneous symmetric polynomial of degree i in ( x , ⋯ , x k ) withthe convention that h i is identically zero for i < and h = .Proof. The exceptional locus of the blowup of X centered at Z is the projective bundle P ( N Z X ) .Let E = c ( O P ( N Z X ) ( )) . By the functoriality properties of Segre classes, we have: f ∗ + E ∩ [ E ] = c ( N Z X ) ∩ [ Z ] = d ∏ i = Z i + Z i , (3.2)where N Z X is the normal bundle of Z in X . The generating function of complete homogeneoussymmetric polynomials in ( x , . . . , x d ) is ∏ dℓ = ( − x ℓ t ) − : ∞ ∑ n = h n ( x , ⋯ , x d ) t n = d ∏ ℓ = − x ℓ t . By matching terms of the same dimensions in equation (3.2), we can compute f ∗ E n in terms ofcomplete homogeneous symmetric polynomials h i ( Z , . . . , Z d ) in the classes Z i : f ∗ E n = ( − ) n − [ t n ] ( d ∏ i = tZ i + tZ i ) = ( − ) d + h n − d ( Z , ⋯ , Z d ) Z ⋯ Z d , where [ t n ] g ( t ) = g n for a formal series g ( t ) = ∑ ∞ i = g i t i and h i is identically zero for i < and h = . Example 3.5. If d = , we have f ∗ E = , f ∗ E = − Z Z , f ∗ E = − ( Z + Z ) Z Z , f ∗ E = − ( Z + Z + Z Z ) Z Z . Example 3.6. If d = , we have f ∗ E = , f ∗ E = , f ∗ E = Z Z Z , f ∗ E = ( Z + Z ) Z Z Z . A direct consequence of Theorem A.2 (Jacobi’s identity) and Lemma 3.4 is the following push-forward formula (see [35]):
Lemma 3.7.
Let Z ⊂ X be the complete intersection of d nonsingular hypersurfaces Z , . . . , Z d meeting transversally in X . Let f ∶ ̃ X Ð→ X be the blowup of X centered at Z with exceptional ivisor E . Then for any integer n ≥ : f ∗ E n = d ∑ ℓ = Z nℓ M ℓ , M ℓ = ∏ m = m ≠ ℓ Z m Z m − Z ℓ . The coefficient M ℓ is the ℓ -moment of the blowup f defined after Theorem 1.8.Proof. f ∗ E n = ( − ) d + h n − d ( Z , ⋯ , Z d ) Z ⋯ Z d (by Lemma 3.4) = ( − ) d + d ∑ ℓ = Z n − ℓ ( d ∏ m = m ≠ ℓ Z ℓ − Z m ) Z ⋯ Z d (by Lemma 1.10) = ( − ) d + d ∑ ℓ = Z nℓ ( d ∏ m = m ≠ ℓ Z m Z ℓ − Z m ) (by the identity Z ⋯ Z d = Z ℓ d ∏ m = m ≠ ℓ Z m ) = d ∑ ℓ = Z nℓ ( d ∏ m = m ≠ ℓ Z m Z m − Z ℓ ) (since d ∏ m = m ≠ ℓ Z m Z ℓ − Z m = ( − ) d − d ∏ m = m ≠ ℓ Z m Z m − Z ℓ )To compute topological invariants of a blowup, we often have to pushforward analytic expressionsof E . Let ̃ Q ( t ) = ∑ a f ∗ Q a t a be a formal power series with Q a ∈ A ∗ ( X ) . The formal series Q ( E ) isa well-defined element of A ∗ ( ̃ X ) . We recall Theorem 1.8: Theorem 1.8.
Let the nonsingular variety Z ⊂ X be a complete intersection of d nonsingularhypersurfaces Z , . . . , Z d meeting transversally in X . Let E be the class of the exceptional divisorof the blowup f ∶ ̃ X Ð→ X centered at Z . Let ̃ Q ( t ) = ∑ a f ∗ Q a t a be a formal power series with Q a ∈ A ∗ ( X ) . We define the associated formal power series Q ( t ) = ∑ a Q a t a whose coefficients pullbackto the coefficients of ̃ Q ( t ) . Then the pushforward f ∗ ̃ Q ( E ) is: f ∗ ̃ Q ( E ) = d ∑ ℓ = Q ( Z ℓ ) M ℓ , where M ℓ = d ∏ m = m ≠ ℓ Z m Z m − Z ℓ . Proof. f ∗ ̃ Q ( E ) = f ∗ ∑ a ( f ∗ Q a ) E a = ∑ a Q a f ∗ E a = ∑ a Q a d ∑ ℓ = Z aℓ M ℓ = ∑ a d ∑ ℓ = Q a Z aℓ M ℓ = d ∑ ℓ = Q ( Z ℓ ) M ℓ . (3.3) We denote the projective bundle of the Weierstrass model to be X = P [ O B ⊕ L ⊗ ⊕ L ⊗ ] and theelliptic fibration ϕ ∶ Y → B to be the zero-scheme of a section of O ( ) ⊗ π ∗ L ⊗ . We denote by19 ( ) the dual of the tautological line bundle of X . We denote by H the first Chern class of O ( ) ,and by L the first Chern class of L . The elliptic fibration ϕ ∶ Y Ð→ B is of class [ Y ] = H + π ∗ L .The classes of the generators of the blowup centers are Z ( n ) i , where n is the number of the blowupmap and i is the number of the center. For example, consider the following blowup: X X X ( x, y, s ∣ e ) ( y, e ∣ e ) (3.4)where each arrow above denotes a blowup, V ( s ) is a smooth divisor in X , and where E n = V ( e n ) isthe exceptional divisor of the n th blowup. The first exceptional divisor is a projective bundle whosefibers have projective coordinates [ x ′ ∶ y ′ ∶ s ′ ] , where x = x ′ e , y = y ′ e , s = s ′ e . For notational convenience, we drop the prime superscripts ( ′ ) appearing after each blowup.The classes associated to the center of the first blowup in (3.4) are: Z ( ) = [ x ] = H + π ∗ L, Z ( ) = [ y ] = H + π ∗ L, Z ( ) = [ s ] = π ∗ S. Likewise, the classes associated to the center of the second blowup are Z ( ) = [ y ] = f ∗ ( H + π ∗ L ) − E , Z ( ) = [ e ] = E . Let us adapt the above data into a matrix-inspired notation, such that i denote columns and n denotes rows. This notation allows us to read the classes of the blowup center by each row. In thisnotation, the above results can be expressed as follows: Z = ⎛⎝ Z ( ) Z ( ) Z ( ) Z ( ) Z ( ) ⎞⎠ = ( H + π ∗ L H + π ∗ L π ∗ Sf ∗ ( H + π ∗ L ) − E E ) . See Table 6 for an exhaustive list of the generator classes associated to the blowup centers of thecrepant resolutions in Table 5. Note that we streamline our notation by omitting the explicit pullbackmaps from the expressions for the classes appearing in these tables.
Using motivitic integration, Kontsevich shows in his famous “String Cohomology” Lecture at Orsaythat birational equivalent Calabi-Yau varieties have the same class in the completed Grothendieckring [47]. Hence, birational equivalent Calabi-Yau varieties have the same Hodge-Deligne polynomial,Hodge numbers, and Euler characteristic. In this section, we compute the Hodge numbers of crepantresolutions of Weierstrass models in the case of Calabi-Yau threefolds.
Theorem 4.1 (Kontsevich, (see [47])) . Let X and Y be birational equivalent Calabi-Yau varietiesover the complex numbers. Then X and Y have the same Hodge numbers. emark 4.2. In Kontsevich’s theorem, a Calabi-Yau variety is a nonsingular complete projectivevariety of dimension d with a trivial canonical divisor. To compute Hodge numbers in this section,we use the following stronger definition of a Calabi-Yau variety. Definition 4.3. A Calabi-Yau variety is a smooth compact projective variety Y of dimension n with a trivial canonical class and such that H i ( Y, O X ) = for ≤ i ≤ n − .We first recall some basic definitions and relevant classical theorems. Definition 4.4.
The
Néron-Severi group NS ( X ) of a variety X is the group of divisors of X moduloalgebraic equivalence. The rank of the Néron-Severi group of X is called the Picard number and isdenoted ρ ( X ) . Theorem 4.5 (Lefschetz ( , ) -theorem, see [72, Theorem 7.2, p. 157] ) . If X is compact Kählermanifold, then the map c ∶ P ic ( X ) → H , ( X, Z ) = H , ( X, C ) ∩ H ( X, Z ) is well-defined and surjec-tive. In addition, the Picard number ρ ( X ) is equal to the Hodge number h , ( X ) ∶ = dim H , ( X, Z ) . Theorem 4.6 (Noether’s formula) . If B is a smooth compact, connected, complex surface withcanonical class K B and Euler number c : χ ( O B ) = − h , ( B ) + h , ( B ) , χ ( O B ) = ( K + c ) . When B is a smooth compact rational surface, we have a simple expression of h , ( B ) as afunction of K using the following lemma. Lemma 4.7.
Let B be a smooth compact rational surface with canonical class K . Then h , ( B ) = − K . (4.1) Proof.
Since B is a rational surface, h , ( B ) = h , ( B ) = . Hence c = + h , ( B ) and the lemmafollows from Noether’s formula.We now compute h , ( Y ) using the Shioda-Tate-Wazir theorem. Theorem 4.8 (Shioda–Tate–Wazir; see [73, Corollary 4.1]) . Let ϕ ∶ Y → B be a smooth ellipticfibration, then ρ ( Y ) = ρ ( B ) + f + rank ( MW ( ϕ )) + where f is the number of geometrically irreducible fibral divisors not touching the zero section. Theorem 4.9.
Let Y be a smooth Calabi-Yau threefold elliptically fibered over a smooth variety B with Mordell-Weil group of rank zero. Then, h , ( Y ) = h , ( B ) + f + , h , ( Y ) = h , ( Y ) − χ ( Y ) , where f is the number of geometrically irreducible fibral divisors not touching the zero section. Inparticular, if Y is a G -model with G a simple group, f is the rank of G . roof. In the statement of the Shioda–Tate–Wazir theorem, we can replace the Picard numbers ρ ( Y ) and ρ ( B ) by the Hodge numbers h , ( Y ) and h , ( B ) using Lefschetz’s (1,1)-theorem. Thatgives h , ( Y ) = h , ( B ) + f + . Since the Euler characteristic of a Calabi-Yau threefold is χ ( Y ) = ( h , − h , ) , and assuming that both χ ( Y ) and h , ( Y ) are known, it follows that h , ( Y ) = h , ( Y ) − χ ( Y ) . Remark 4.10.
For G -models with G a simple group, f will be the rank of G [58, §4]. In this section, we discuss in detail the computation of the Euler characteristic of SU(2)-models.Note that the results presented in this section are equivalent for each of the four possible Kodairafibers (namely, types I s , I s , I s , III, IV ns ) realizing an SU(2)-model; see Section 6 for a list of theWeierstrass equations defining the various SU(2)-models. We find c ( X ) = ( + H )( + H + π ∗ L )( + H + π ∗ L ) c ( B ) c ( Y ) = ( H + π ∗ L ) c ( X ) + H + π ∗ L .
The singular elliptic fibration is resolved by a unique blowup with center ( x, y, s ) [31]. We denotethe blowup by f ∶ X Ð→ X and the exceptional divisor by E . The center is a complete intersectionof hypersurfaces V ( x ) , V ( y ) , and V ( s ) , whose classes are respectively Z = π ∗ L + H, Z = π ∗ L + H, Z = π ∗ S. The proper transform of the elliptic fibration Y is denoted Y , and is obtained from the totaltransform of Y by removing E . It follows that the class of Y in X is [ Y ] = [ f ∗ ( H + π ∗ L ) − E ] ∩ [ X ] Moreover, we have the following Chern classes: c ( T X ) = ( + E ) ( + f ∗ Z − E )( + f ∗ Z − E )( + f ∗ Z − E )( + f ∗ Z )( + f ∗ Z )( + f ∗ Z ) f ∗ c ( T X ) c ( T Y ) = ( + E )( + f ∗ Z − E )( + f ∗ Z − E )( + f ∗ Z − E )( + H + L − E )( + f ∗ Z )( + f ∗ Z )( + f ∗ Z ) f ∗ c ( T X ) By an expansion of c ( T Y ) in first order, we can easily check that the resolution is crepant: c ( T Y ) = f ∗ c ( T Y ) . After the blowup, the homological total Chern class is c ( Y ) = c ( T Y ) ∩ [ Y ] : c ( Y ) = ( f ∗ H + f ∗ π ∗ L − E )( + E ) ( + f ∗ Z − E )( + f ∗ Z − E )( + f ∗ Z − E )( + f ∗ Z )( + f ∗ Z )( + f ∗ Z ) f ∗ c ( X ) .
22o compute the Euler characteristic, we have to evaluate χ ( Y ) = ∫ Y c ( Y ) . The first pushforward requires the following data: M = Z Z ( Z − Z )( Z − Z ) , M = Z Z ( Z − Z )( Z − Z ) , M = Z Z ( Z − Z )( Z − Z ) . Applying the pushforward theorem is now a purely algebraic routine that can be easily implementedin one’s favorite algebraic software. Using Theorem 1.8, we pushforward c ( Y ) from the Chow ring A ∗ ( X ) to the Chow ring A ∗ ( X ) . Using Theorem 1.11, we then pushforward f ∗ c ( Y ) to the Chowring of the base. When the dust settles, we find an expression of χ ( Y ) in the Chow ring of the base: χ ( Y ) = ∫ Y c ( T Y ) = ∫ X f ∗ c ( T Y ) = ∫ B π ∗ f ∗ c ( T Y ) = ∫ B L + LS − S ( + S )( + L − S ) c ( T B ) . Concretely, we replace c ( T B ) by the Chern polynomial c t ( T B ) = + c t + c t + c t + ⋯ , L by Lt ,and S by St ; if d is the dimension of B , the Euler characteristic of Y is given by the coefficient of t d in the Taylor expansion centered at t = of the generating function: χ ( Y ) = Lt + LSt − S t ( + St )( + Lt − St ) c t ( T B ) = Lt + t ( c L − L + LS − S ) ++ t ( − c L + c LS − c S + c L + L − L S + LS − S ) + ⋯ Theorem 5.1. If B is a curve, the Euler characteristic of an SU(2) model is L . If B is a surface,the Euler characteristic is ( c L − L + LS − S ) . If B is a threefold, the Euler characteristic is ( − c L + c LS − c S + c L + L − L S + LS − S ) . In order to consider the Calabi-Yau case, we set L = c ( T B ) in the above expression, which gives χ ( Y ) = c t − t ( c − c S + S ) + t ( c − c S + c c + c S − S ) + ⋯ Note that we retrieve the result for a smooth Weierstrass model if we further impose S = . Remark 5.2.
As a byproduct of the computation of the Euler characteristic of the resolution, wecan also easily evaluate the contribution from the singularities to be L + LS − S ( + S )( + L − S ) c ( T B ) − L + L c ( T B ) = ( L + L − LS − S ) S ( + L )( + L − S )( + S ) c ( T B ) , which can be rewritten as χ ( Y ) − χ ( Y ) = L − LS + L − S ( + L )( + L − S ) c ( S ) , c ( S ) = S + S c ( T B ) ∩ [ B ] . In the Calabi-Yau case L = c ( T B ) , the above quantity usually has a physical meaning. For example,23f Y is a Calabi-Yau fourfold, this expression reduces to − S ( c − S ) ∩ [ B ] , which is the contributionof branes to the Euler characteristic. In another limit, the above expression can be understood asthe contribution of the G -flux in M-theory to the M2-brane flux or brane flux in type IIB stringtheory: ∫ Y G ∧ G = ∫ S F ∧ F = − ∫ S ( c − S ) . The G -models studied in this paper are all realized as crepant resolutions of the singular Weierstrassmodel y z + a xyz + a yz − ( x + a x z + a xz + a z ) = , where the desired singularity structures corresponding to the decorated Kodaira fibers can be spec-ified by the valuation of the coefficients of the Weierstrass equation with respect to the divisor S = V ( s ) . Following Tate’s algorithm, we use the notation a i,p = a i / s p , where the valuations p arethe minimal values dictated by Tate’s algorithm and we assume that the coefficients a i,p are generic.We present the results of our computation of the Euler characteristic generating functions forvarious G -models. The generating functions are the pushforwards of the homological total Chernclass of the resolved Weierstrass model to the base B , and are expressed as rational functions of theclasses S and L (where L = c ( L ) is the class of the fundamental line bundle and S is the class of thedivisor in the base B ), multiplied by the total Chern class of the base, c ( B ) —see Table 7. Tables 8-10specialize the results to (respectively) elliptic threefolds, fourfolds, and elliptic Calabi-Yau fourfolds,while Table 11 summarizes the Hodge numbers for Calabi-Yau threefold G -models.When computing Hodge numbers of a G model which is a Calabi-Yau threefold, we recall that weassume that the base is a rational surface. This is a direct consequence of Definition 4.3. Moreover,for a G -model with G a simple group, the integer f that enters in Theorem 4.9 is the rank of G .For the SO(3), SO(5), and SO(6)-models, the class S is given by [28]: ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ S = L for SO(3) ,S = L for SO(5) ,S = L for SO(6) . Below we list the various Weierstrass equations we use to compute the G -models, labeled by theirKodaira fiber type and associated Lie group G . It is necessary to specify a crepant resolution inorder to actually compute the total Chern class and Euler characteristic of a G -model. There couldbe several distinct crepant resolutions for a G -model. However, Theorem 2.8 assures that the Eulercharacteristic is insensitive to the choice of crepant resolution and therefore we only need one crepantresolution to compute the Euler characteristic of a G -model defined by the crepant resolution of a24eierstrass model. The models associated to the groups SU( n ) and USp( n ) are [45]:I s SU(2) ∶ y z + a xyz + a , syz = x + a , sx z + a , sxz + a , s z , (6.1)I ns n USp( n ) ∶ y z = x + a x z + a ,n s n xz + a , n s n z , (6.2)I ns n + USp( n ) ∶ y z = x + a x z + a ,n + s n + xz + a , n + s n + z , (6.3)I s n SU( n ) ∶ y z + a xyz = x + a , sx z + a ,n s n xz + a , n s n z , (6.4)I s n + SU( n + ) ∶ y z + a xyz + a ,n s n yz = x + a , sx z + a ,n + s n + xz + a , n + s n + z . (6.5)The Weierstrass models for SO(3), SO(5), and SO(6) are discussed in [28]; these models require aMordell-Weil group Z / Z . The crepant resolutions of the Weierstrass models for G , Spin(7), andSpin(8) models are studied in [27] and require a careful analysis of the Galois group of an associatedpolynomial. The Weierstrass equations defining these models along with the remaining G -models,with G one of the exceptional groups are given below [45, 28]:I ns SO(3) ∶ y z = x ( x + a xz + a z ) , (6.6)I ns SO(5) ∶ y z = ( x + a x z + s xz ) , (6.7)I s SO(6) ∶ y z + a xyz = x + msx z + s xz , m ∈ C , m ≠ − , , , (6.8)I ∗ ss Spin(7) ∶ y z = x + a , sx z + a , s xz + a , s z , (6.9)I ∗ s Spin(8) ∶ y z = ( x − x sz )( x − x sz )( x − x sz ) + s rx z + s qxz + s tz , (6.10)III SU(2) ∶ y z = x + sa , xz + s a , z , (6.11)IV ns SU(2) ∶ y z = x + s a , xz + s a , z , (6.12)IV s SU(3) ∶ y z + a , syz = x + s a , xz + s a , z , (6.13)I ∗ ns G ∶ y z = x + s a , xz + s a , z , (6.14)IV ∗ ns F ∶ y z = x + s a , xz + s a , z , (6.15)IV ∗ s E ∶ y z + a , s yz = x + s a , xz + s a , z , (6.16)III ∗ E ∶ y z = x + s a , xz + s a , z , (6.17)II ∗ E ∶ y z = x + s a , xz + s a , z . (6.18) Theorem 6.1.
Let Y → B be a singular Weierstrass model of a G -model. If f ∶ Y → Y is a crepantresolution of Y given by one of the sequence of blowups given in Table 5, the generating function ofthe Euler characteristic of any crepant resolution of Y is given by the corresponding entry in Table7. Remark 6.2.
The theorem does not address if the sequence of blowups define a crepant resolution.One usually has to assume some conditions on the coefficients of the Weierstrass equations. See forexample [27]. In some cases, the dimension of the base plays a role too [27].25 roup Fiber Type Crepant ResolutionSU ( ) I s , I ns I ns , IIIIV ns X X ( x, y, s ∣ e ) SU ( ) USp ( ) G I s , IV s I ns I ∗ ns X X X ( x, y, s ∣ e ) ( y, e ∣ e ) SU ( ) Spin ( ) I s I ∗ ss X X X X ( x, y, s ∣ e ) ( y, e ∣ e ) ( x, e ∣ e ) Spin ( ) I ∗ s X X X X X ( x, y, s ∣ e ) ( y, e ∣ e ) ( x − x i sz, e ∣ e ) ( x − x j sz, e ∣ e ) F IV ∗ ns X X X X X ( x, y, s ∣ e ) ( y, e ∣ e ) ( x, e ∣ e ) ( e , e ∣ e ) SU ( ) I s X X X X X ( x, y, s ∣ e ) ( x, y, e ∣ e ) ( y, e ∣ e ) ( y, e ∣ e ) Spin ( ) I ∗ s X X X X X X ( x, y, s ∣ e ) ( y, e ∣ e ) ( x, e ∣ e ) ( y, e ∣ e ) ( e , e ∣ e ) E IV ∗ s X X X X X X X ( x, y, s ∣ e ) ( y, e ∣ e ) ( x, e ∣ e ) ( e , e ∣ e ) ( y, e ∣ e )( y, e ∣ e ) E III ∗ X X X X X X X X ( x, y, s ∣ e ) ( y, e ∣ e ) ( x, e ∣ e ) ( y, e ∣ e ) ( e , e ∣ e )( e , e ∣ e )( e , e ∣ e ) E II ∗ X X X X X X X X X ( x, y, s ∣ e ) ( y, e ∣ e ) ( x, e ∣ e ) ( y, e ∣ e ) ( e , e ∣ e )( e , e ∣ e )( e , e , e ∣ e )( e , e ∣ e ) SO ( ) I ns X X ( x, y ∣ e ) SO ( ) I ns X X X ( x, y, s ∣ e ) ( x, y, e ∣ e ) SO ( ) I s X X X X ( x, y, s ∣ e ) ( y, e ∣ e ) ( x, e ∣ e ) Table 5: The blowup centers of the crepant resolutions. See the beginning of Section 3.2 for anexplanation of our notation. 26 lgebra Group Generator classes of the blowup centers ( Z ( n ) i ) A SU ( ) ( H + L H + L S ) A C G SU ( ) USp ( ) G ( H + L H + L SH + L − E E ) A SU ( ) Spin ( ) ⎛⎜⎝ H + L H + L SH + L − E E H + L − E E ⎞⎟⎠ D Spin ( ) ⎛⎜⎜⎜⎝ H + L H + L SH + L − E E H + L − E E H + L − E E − E ⎞⎟⎟⎟⎠ F F ⎛⎜⎜⎜⎝ H + L H + L SH + L − E E H + L − E E E − E E ⎞⎟⎟⎟⎠ A SU ( ) ⎛⎜⎜⎜⎝ H + L H + L SH + L − E H + L − E E H + L − E − E E − E H + L − E − E − E E ⎞⎟⎟⎟⎠ D Spin ( ) ⎛⎜⎜⎜⎜⎜⎜⎝ H + L H + L SH + L − E E H + L − E E H + L − E − E E E − E E − E ⎞⎟⎟⎟⎟⎟⎟⎠ E E ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝ H + L H + L SH + L − E E H + L − E E E − E E H + L − E − E E − E H + L − E − E − E E ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠ E E ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ H + L H + L SH + L − E E H + L − E E H + L − E − E E E − E E − E E − E − E E E − E E ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ E E ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ H + L H + L SH + L − E E H + L − E E H + L − E − E E E − E E − E E E E − E − E E − E E E − E − E E ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ A SO ( ) ( H + L H + L ) B SO ( ) ( H + L H + L LH + L − E H + L − E E ) A SO ( ) same as SU(4), with S = L Table 6: The classes of the centers of the blowups for all G -models27lgebra Group Kodaira Fiber χ ( Y ) = π ∗ ( f ∗ c ( T Y ) ∩ [ Y ]) − { e } I L + L c ( B ) A SU ( ) I s , I ns I ns , IIIIV ns L + LS − S ( + S )( + L − S ) c ( B ) A SU ( ) I s , IV s C USp ( ) I ns L + SL − S ( + S )( + L − S ) c ( B ) G G I ∗ ns A B SU ( ) Spin ( ) I s I ∗ ss L + L + LS − S + L S − LS + S ( + S )( + L − S )( + L − S ) c ( B ) D F Spin ( ) F I ∗ s IV ∗ ns L + SL − S ( + S )( + L − S ) c ( B ) A SU ( ) I s L + L S + L − LS + LS − S ( + L )( + S )( + L − S ) c ( B ) D Spin ( ) I ∗ s ( − ( L + ) S + ( L + ) LS + ( L + ) L + S )( S + )( − L + S − )( − L + S − ) c ( B ) E E IV ∗ s L + L − S + SL − S L + SL + S ( + S )( + L − S )( + L − S ) c ( B ) E E III ∗ L + L + LS − S + L S − LS + S ( + S )( + L − S )( + L − S ) c ( B ) E E II ∗ L + LS − S ( + S )( + L − S ) c ( B ) A SO ( ) I ns L + L c ( B ) B SO ( ) I ns L ( + L )( + L ) c ( B ) A SO ( ) I s L + L c ( B ) Table 7: Generating functions of Euler characteristic of crepant resolutions of Tate’s models withtrivial Mordell-Weil groups. S is the divisor over which the generic fiber is of type given by theKodaira fiber and L = c ( L ) where L is the fundamental line bundle of the Weierstrass model.28 odels χ ( Y ) , Euler characteristicSmooth Weierstrass L ( c − L ) SU ( ) ( c L − L + LS − S ) SU ( ) or USp ( ) or G ( c L − L + LS − S ) SU ( ) or Spin ( ) ( c L − L + LS − S ) Spin ( ) or F ( c L − L + LS − S ) SU ( ) ( c L − L + LS − S ) Spin ( ) ( c L − L + LS − S ) E ( c L − L + LS − S ) E ( c L − L + LS − S ) E ( c L − L + LS − S ) SO ( ) L ( c − L ) SO ( ) L ( c − L ) SO ( ) L ( c − L ) Table 8: Euler characteristic for elliptic threefolds
Models χ ( Y ) , Euler characteristicSmooth Weierstrass L ( − c L + c + L ) SU ( ) ( − c L + c LS − c S + c L + L − L S + LS − S ) SU ( ) or USp ( ) or G ( − c L + c LS − c S + c L + L − L S + LS − S ) SU ( ) or Spin ( ) ( − c L + c LS − c S + c L + L − L S + LS − S ) SU ( ) − c L + c LS − c S + c L + L − L S + LS − S Spin ( ) ( − c L + c LS − c S + c L + L − L S + LS − S ) Spin ( ) or F ( − c L + c L + L + c LS − c S − L S + LS − S ) E ( − c L + c LS − c S + c L + L − L S + LS − S ) E ( − c L + c LS − c S + c L + L − L S + LS − S ) E ( − c L + c LS − c S + c L + L − L S + LS − S ) SO ( ) L ( L − c L + c ) SO ( ) L ( L − c L + c ) SO ( ) L ( L − Lc + c ) Table 9: Euler characteristic for elliptic fourfolds
Models χ ( Y ) , Euler characteristicSmooth Weierstrass c c + c SU ( ) ( c c + c − c S + c S − S ) SU ( ) or USp ( ) or G ( c c + c − c S + c S − S ) SU ( ) or Spin ( ) ( c c + c − c S + c S − S )) Spin ( ) or F ( c c + c − c S + c S − S ) SU ( ) ( c c + c − c S + c S − S ) Spin ( ) ( c c + c − c S + c S − S ) E ( c c + c − c S + c S − S ) E ( c c + c − c S + c S − S ) E ( c c + c − c S + c S − S ) SO ( ) c ( c + c ) SO ( ) c ( c + c ) SO ( ) c ( c + c ) Table 10: Euler characteristic for Calabi-Yau elliptic fourfolds where c = L .29 lgebra Group Kodaira Fiber h , ( Y ) h , ( Y ) χ ( Y ) − { e } I − K + K − K A SU ( ) I s , I ns I ns , IIIIV ns − K + K + KS + S − K − KS − S A SU ( ) I s , IV s C USp ( ) I ns − K + K + KS + S − K − KS − S G G I ∗ ns A SU ( ) I s − K + K + KS + S − K − KS − S B Spin ( ) I ∗ ss D Spin ( ) I ∗ s − K + K + KS + S − K − KS − S F F IV ∗ ns A SU ( ) I s − K + K + KS + S − K − KS − S D Spin ( ) I ∗ s − K + K + KS + S − K − KS − S E E IV ∗ s − K + K + KS + S − K − KS − S E E III ∗ − K + K + KS + S − K − KS − S E E II ∗ − K + K + KS + S − K − KS − S A SO ( ) I ns − K + K − K B SO ( ) I ns − K + K − K A SO ( ) I s − K + K − K Table 11: Hodge numbers and Euler characteristic of Calabi-Yau threefolds obtained from crepantresolutions of Tate’s models. 30
Discussion
In this paper, we have computed the generating functions for the Euler characteristics of G -modelsobtained by crepant resolutions of Weierstrass models with bases of arbitrary dimension. The caseof G -models that are also Calabi-Yau varieties is important in string theory and is treated here asa special case. In particular, we list the Euler characteristic of G -models that are elliptic threefoldsand fourfolds. For Calabi-Yau threefolds, we also compute the Hodge numbers. These results areinsensitive to the particular choice of resolution due to Batyrev’s theorem on the Betti numbers ofcrepant birational equivalent varieties and Kontsevich’s theorem on the Hodge numbers of birationalequivalent Calabi-Yau varieties [6, 47]. We have considered all possible G -models with G a simpleLie group, except for the case of Kodaira fibers I n > and I ∗ n > that we will treat in a follow-up paper.We start with a G -model given by a singular Weierstrass model ϕ ∶ Y Ð→ B with a fundamentalline bundle L (in the Calabi-Yau case, c ( L ) = c ( T B ) ). Given a crepant resolution f ∶ Y Ð→ Y determined by a sequence of blowups with smooth centers that are complete intersections withnormal crossings, we compute the Euler characteristic of Y as the degree of its total Chern classdefined in homology χ ( Y ) = ∫ Y c ( Y ) . We work relative to a smooth base B of arbitrary dimension. Using the functorial properties of thedegree, we pushforward first to the Chow ring of the projective bundle and then to the Chow ringof the base: χ ( Y ) = ∫ B π ∗ f ∗ c ( Y ) . The final result is a generating function for the Euler characteristic.A key result of this work is Theorem 1.8, which has numerous applications in intersection theory.We also provide a simple proof of an identity (Lemma 1.10) that can be traced back to Jacobi’sthesis and appears in numerous situations in mathematics and physics, which is instrumental in theproof of Theorem 1.8.We also retrieve in a unifying way known results on the Euler characteristics and Hodge numbersof Calabi-Yau threefolds. Furthermore, we have proven en passant a conjecture of Blumenhagen,Grimm, Jurke, and Weigand [9] on the Euler characteristics of Calabi-Yau fourfolds that are G -models with G belonging to the exceptional series. One interesting point that is almost trivial fromthe perspective taken in this paper is that certain G -models with different G will have the sameEuler characteristic just because they are resolved by the same sequence of blowups. Acknowledgements
The authors are grateful to Paolo Aluffi, Jim Halverson, Remke Kloosterman, Cody Long, KenjiMatsuki, Julian Salazar, Shu-Heng Shao, and Shing-Tung Yau for helpful discussions. The authorswould like in particular to acknowledge Andrea Cattaneo for many useful comments and suggestions.The authors are thankful to all the participants of the workshop “A Three-Workshop Series on theMathematics and Physics of F-theory” supported by the National Science Foundation (NSF) grantDMS-1603247. M.E. is supported in part by the National Science Foundation (NSF) grant DMS-31701635 “Elliptic Fibrations and String Theory”. P.J. is supported by NSF grant PHY-1067976. P.J.would like to extend his gratitude to Cumrun Vafa for his tutelage and continued support. M.J.K.would like to acknowledge partial support from NSF grant PHY-1352084. M.J.K. is thankful toDaniel Jafferis for his guidance and constant support.
A Jacobi’s Partial Fraction Identity
In this section, we prove a formula of Jacobi and exploit the theorem to give a simple proof of aformula of Louck and Biedenharn [51, Appendix A, p. 2400] by demonstrating its equivalence withthe following theorem of Jacobi.
Theorem A.1 (Jacobi, [43, Section III.17, p. 29-30]) . Let a i ( i = , . . . , d ) be d distinct elements ofan integral domain. Then d ∏ i = x − a i = d ∑ i = x − a i d ∏ j = j ≠ i a i − a j . (A.1) Proof.
Let F ( x ) = d ∏ i = x − a i , (A.2)where a i ≠ a j for i ≠ j . We would like to find the partial fraction expansion of F ( x ) . That is, wewould like to find coefficients A i ( i = , ⋯ , d ) such that F ( x ) = d ∑ i = A i x − a i . (A.3)We determine A i by the method of residues . Multiplying (A.3) by ( x − a i ) , simplifying, and evaluatingat x = a i gives ( x − a j ) F ( x )∣ x = a j = A j . Applying the above formula to (A.2), we get A j = ∏ i ≠ j a i − a j , which is the identity of Jacobi: d ∏ i = x − a i = d ∑ i = x − a i d ∏ j = j ≠ i a i − a j . (A.4) Theorem A.2 (Jacobi, Louck–Biedenharn, Cornelius) . Let h r ( x , ⋯ , x d ) be the homogeneous com-plete symmetric polynomial of degree r in d variables of an integral domain. Then, h r ( x , ⋯ , x d ) = d ∑ ℓ = x r + d − ℓ d ∏ m = m ≠ ℓ x ℓ − x m . This theorem was proven by Louck-Biedenharn [51, Appendix A, p. 2400] and Cornelius [15]. We32resent a new and much simpler proof below by showing that the theorem is simply a reformulationof Jacobi’s identity (Theorem A.1).
Proof.
Substituting x → / t in Equation (A.1) gives: d ∏ i = t − a i t = d ∑ i = t − a i t d ∏ j = j ≠ i a i − a j . Expanding /( − a i t ) in both side of the equation gives t d ∞ ∑ r = h r ( a , . . . , a d ) t r = t d ∑ i = ∞ ∑ k = a ki t k d ∏ j = j ≠ i a i − a j ∞ ∑ r = h r ( a , . . . , a d ) t r + d − = ∞ ∑ k = ( d ∑ i = a ki d ∏ j = j ≠ i a i − a j ) t k . Comparing terms of the same degree in t , we get the final expression of Lemma 1.10: h r ( a , . . . , a d ) = d ∑ i = a r + d − i d ∏ j = j ≠ i a i − a j . B The Euler Characteristic as the Degree of the Top Chern Class
The purpose of this section is to explain from different points of view why the Euler characteristicis the degree of the top Chern class. Traditionally, this statement is seen as a generalization ofthe Poincaré–Hopf theorem that asserts that the total degree of a vector field defined on a smoothmanifold M is the Euler characteristic of M . This statement can also be seen as a generalizationof the Gauss–Bonnet–Chern Theorem (which is itself is a consequence of Poincaré–Hopf theorem).Here we will review three different approaches. The first one relies on Leftschetz fixed point theorem.The second one uses he Poincaré–Hopf theorem using the interpretation of Chern classes as relatedto the class of some degenerated loci as discussed in Chapter 3 of Fulton. The third one is anapplication of the Hirzebruch–Riemann–Roch theorem and the Hodge decomposition theorem.Let M be a smooth compact manifold. The k th Betti number of M is by definition the dimensionof the cohomology group H k ( M, Q ) . The Euler characteristic of M is denoted by χ ( X ) and is definedas the following alternative sum of Betti numbers of M : χ ( M ) ∶ = dim M ∑ k = ( − ) k b k , b k ∶ = dim H i ( M, Q ) . .1 Lefschetz fixed point theorem and the Euler characteristic as an intersectionnumber Theorem B.1 (Lefschetz fixed point theorem) . Let M be a compact smooth manifold of dimension m and f ∶ M Ð→ M a continuous map. We define the Lefschetz number of f as L ( f ) ∶ = m ∑ k = ( − ) k tr ( f ∗ ∣ H k ( M, Q )) , f ∗ ∶ H k ( M, Q ) Ð→ H k ( M, Q ) . Then L ( f ) is equal to the intersection number of the graph Γ f of f and the diagonal ∆ in M × ML ( f ) = ∫ M × M Γ f ⋅ ∆ . Thus, the Leftschetz number L ( f ) is the number of fixed points of f counted with multiplicities. Corollary.
Let M be a compact smooth manifold and ∆ be the diagonal of M × M , then the Eulercharacteristic of M , χ ( M ) = ∑ i ( − ) k dim H i ( M, Q ) , is equal to the self-intersection of ∆ in M × M : χ ( M ) = ∫ M × M ∆ ⋅ ∆ . Proof.
Consider the special case of Lefschetz theorem for which f is the identify map on M . Then,the Leftschetz number reduces to the Euler characteristic χ ( M ) as the trace tr ( f ∗ ∣ H k ( M, Q )) becomes the k th Betti number b k of M and the intersection number ∫ M × M Γ f ⋅ ∆ becomes theself-intersection of the diagonal ∆ in M × M . Theorem B.2 (Self-intersection formula, see [36, Corollary 6.3, p. 102-103]) . Let i ∶ Z → X be aregular imbedding of codimension d and normal bundle N . Then for any α ∈ A ∗ ( Z ) we have theself-intersection formula i ∗ i ∗ ( α ) = c d ( N ) ∩ α. Theorem B.3. If X is a nonsingular complete algebraic variety, then the Euler characteristic of X is equal to the degree of the total homological Chern class of X : χ ( X ) = ∫ c ( X ) , c ( X ) ∶ = c ( T X ) ∩ [ X ] . Proof.
The theorem follows from the previous corollary expressing the Euler characteristic χ ( X ) asthe self-intersection of the diagonal ∆ in X × X , followed by the self-intersection formula expressing ∆ ⋅ ∆ as the class c dim X ( N ∆ X × X ) ∩ [ ∆ ] . Since the normal bundle of ∆ in X × X is isomorphic to thetangent bundle of X (see for example [12, Lemma 11.23, p. 127]), it follows that [36, Example 8.1.12,p. 136], the self-intersection of the diagonal ∆ in X × X is ∫ c dim X ( T X ) ∩ [ X ] = ∫ c ( T X ) ∩ [ X ] : χ ( X ) = ∫ X × X ∆ ⋅ ∆ = ∫ c ( N ∆ X × X ) ∩ [ ∆ ] = ∫ c ( T X ) ∩ [ X ] . .2 Poincaré-Hopf theorem and the Euler characteristic Theorem B.4 (Poincaré-Hopf) . Let M be a smooth compact manifold without boundary and v bea vector field with isolated zeros. Then the sum of the local indices at the zeros of v is equal to theEuler characteristic of M . Remark B.5.
This theorem can be generalized to manifolds with boundaries by requiring v topoint outward. Poincaré proved a two dimensional version of this theorem in 1885. The generalversion was proven by Hopf in 1926. Theorem B.6 ([36, Example 3.2.16, p. 61]) . Let E be a vector bundle of rank r on a smooth variety X , let s be a section of E , and Z the zero-scheme of s . If X is purely n -dimensional and s is aregular section, then Z is purely ( n − r ) -dimensional, and [ Z ] = c r ( E ) ∩ [ X ] . In particular, if E is the tangent bundle T X of X , then r (i.e. the rank of E ) is the dimension of X , and the section s of E is just a vector field. The zero-scheme Z is a -cycle that is the sum of theisolated singularities of s counted with multiplicities. Hence, the degree of the top Chern class of T X gives the index of the vector field s , which is the Euler characteristic of M by the Poincaré–Hopftheorem. Since the degree of c ( X ) is exactly the degree of c r ( T X ) ∩ [ X ] , we retrieve Theorem B.3: χ ( X ) = ∫ c ( X ) . B.3 Hirzebruch–Riemann–Roch theorem and the Euler characteristic
In this sub-section, using the Hirzebruch–Riemann–Roch theorem and the Hodge decompositiontheorem, we prove that the Euler characteristic of a nonsingular projective variety is the degree ofits homological total Chern class. We follow Fulton ([36, Example 18.3.7, p. 362] and [36, Example3.2.5, p. 57]) as presented by D. Rössler [65]. We denote the Todd class, the Chern character, andthe dual of a vector bundle E by td ( E ) and ch ( E ) , and E ∨ respectively.Let X be a projective variety of dimension d and V a coherent sheaf defined over X . We denoteby H q ( X, V ) the q -th cohomology group of X with coefficients in the sheaf of germs of local sectionsof V . The cohomology groups H q ( X, V ) vanish for q > d and are all finite dimensional for ≤ q ≤ d .The Euler characteristic of V in X is by definition the finite number χ ( X, V ) ∶ = d ∑ q = ( − ) q dim H q ( X, V ) . The Hirzebruch–Riemann–Roch theorem provides an expression for χ ( X, V ) in terms of character-istic classes of T X and V realizing a conjecture of Serre in a letter to Kodaira and Spencer. Theorem B.7 (Hirzebruch–Riemann–Roch) . Let V be a coherent sheaf over a nonsingular variety X . Then χ ( X, V ) = ∫ X ch ( V ) td ( T X ) .
35e will also need the following lemma relating the Todd class and the Chern character. Thislemma is instrumental in the proof of the Hirzebruch–Riemann–Roch theorem of Borel and Serre[11, Lemma 18, p. 128], and is also discussed by Fulton in [36, Example 3.2.5, p. 57].
Lemma B.8 (Hirzebruch, [40, Theorem 10.1.1, page 92])) . Let E be a vector bundle of rank r .Then ch ( r ∑ q = ( − ) q ⋀ q E ∨ ) td ( E ) = c r ( E ) . Proof.
By the splitting principal, we can always formally factorize the total Chern class of E as c ( E ) = ∏ i ( + a i ) , where a i are the Chern roots of E . Then by definition ch ( E ) ∶ = r ∑ i = e a i , td ( E ) ∶ = ∏ i a i ( − e − a i ) . We have the classical relations (see [40, Theorem 4.4.3, page 64] or [36, Remark 3.2.3, p. 54–56]) c ( E ∨ ) = ∏ i ( − a i ) , c ( ⋀ q E ) = ∏ ≤ i <⋯< i q ≤ r ( + a i + ⋯ + a i q ) Hence ch ( ⋀ q E ∨ ) = ∑ ≤ i <⋯< i q ≤ r e −( a i +⋯+ a iq ) Thus by the additive properties of the Chern character and the definition of the Todd class: ch ( r ∑ q = ( − ) q ⋀ q E ∨ ) = r ∑ q = ( − ) q ch ( ⋀ q E ∨ ) = r ∏ i = ( − e − a i ) = ( a . . . a r ) r ∏ i = ( − e − a i ) a i = c r ( E ) td − ( E ) . Theorem B.9.
Let X be a nonsingular complete projective variety defined over the complex numbers.Then the Euler characteristic χ ( X ) = ∫ c ( X ) . Proof.
For X a nonsingular variety of dimension d , we apply lemma [ ? ] to the tangent bundle E = T X and we note that E ∨ = T X ∨ ∶ = Ω X , where Ω X is the sheaf of differentials of X , and bydefinition, the sheaf of differential p -forms is ⋀ q Ω X ∶ = Ω qX . Hence, we get ch ( d ∑ q = ( − ) q Ω qX ) td ( T X ) = c r ( T X ) .
36e rewrite the left hand side of the previous equation as follows ∫ X ch ( d ∑ q = ( − ) q Ω qX ) td ( T X ) = d ∑ q = ( − ) q ∫ X ch ( Ω qX ) td ( T X ) = d ∑ q = ( − ) q ∫ X χ ( X, Ω qX ) = d ∑ q = d ∑ p = ( − ) p + q dim H p ( X , Ω qX ) = d ∑ k = ( − ) k ∑ p + q = k dim H p ( X , Ω qX ) = d ∑ k = ( − ) k b k = χ ( X ) . The first equality is a direct consequence of the additive property of the Chern character, the secondequality is due to the Hirzebruch–Riemann-Roch theorem applied to Ω qX , the third equality followsfrom the definition of the Euler characteristic of a sheaf, and the fifth equality is a direct applicationof the Hodge decomposition theorem Ω k = ⊕ p + q = k Ω p,q and Dolbeault’s theorem, which asserts thatthe Dolbeault cohomology is isomorphic to the sheaf cohomology of the sheaf of differential forms: H p,q ( X ) ≅ H p ( X, Ω qX ) . In particular, h p,q ( X ) = dim H p ( X , Ω qX ) are the Hodge numbers of X . Thelast equality is by the definition of the Euler characteristic. Hence, since ∫ c ( X ) = ∫ c ( T X ) ∩ [ X ] = ∫ X c r ( T X ) , we get ∫ c ( X ) = χ ( X ) . C Basic Notions
The local ring of a subvariety S of X is denoted O X,S , its maximal ideal is M X,S and the quotientfield is the residue field κ ( S ) = O X,S / M X,S . The local ring O X,S is the stalk of the structure sheafof X at the generic point η S of S and κ ( S ) is the function field of S . If S is a divisor, O X,S is a onedimensional local domain. In case X is nonsingular along S , O X,S is a discrete valuation ring andthe order of vanishing is given by the usual valuation.
C.1 Fiber types, dual graphs, Kodaira symbols
Definition C.1 (Algebraic cycle) . An algebraic cycle of a Noetherian scheme X is a finite formalsum ∑ i N i V i of subvarieties V i with integer coefficients N i . If all the subvarieties V i have the samedimension d , the cycle is called a d -cycle. The free group generated by subvarieties of dimension d is denoted Z d ( X ) . The group of all cycles, denoted Z ( X ) = ⊕ d Z d ( X ) , is the free group generatedby subvarieties of X . 37 efinition C.2 (Degree of a zero-cycle[36, Chapter 1, Definition 1.4, p. 13]) . Let X be a completescheme. The degree of a zero-cycle ∑ N i p i of X is deg ( ∑ i N i p i ) = ∑ i N i [ κ ( p i ) ∶ k ] , where [ κ ( p i ) ∶ k ] is the degree of the field extension κ ( p i ) → k .Let Θ be an algebraic one-cycle with irreducible decomposition Θ = ∑ i m i Θ i . We denote by Θ i ⋅ Θ j the zero-cycle defined by the intersection of Θ i and Θ j for i ≠ j . Definition C.3 ( n -points, tree) . A n -point of an algebraic one-cycle Θ is a point in ⋃ i Θ i , whichbelongs to exactly n distinct irreducible components Θ i . An algebraic one-cycle Θ is said to be a tree if it does not have n -points for n > . Two curves intersect transversally if their intersectionconsists of isolated reduced closed points.Following Kodaira [46], we introduce the following definition: Definition C.4 (Fiber type) . By the type of an algebraic one-cycle Θ ∈ Z ( X ) with irreducibledecomposition Θ = ∑ i m i Θ i , we mean the isomorphism class of each irreducible curve Θ i , togetherwith the topological structure of the reduced polyhedron ∑ Θ i (that is the collection of zero-cycles Θ i ⋅ Θ j ( i ≠ j )), and the homology class of Θ = ∑ i m i Θ i in the Chow group A ( X ) . Example C.5.
For instance, Θ ⋅ Θ = p + p indicates that the two curves Θ and Θ meet attwo points p and p with respective intersection multiplicity and . Definition C.6 (Dual graph) . To an algebraic one-cycle Θ with irreducible decomposition Θ = ∑ i m i Θ i , we associate a weighted graph (called the dual graph of Θ ) such that:• The vertices are the irreducible components of the fiber.• The weight of a vertex corresponding to the irreducible component Θ i is its multiplicity m i .When the multiplicity is one, it can be omitted.• The vertices corresponding to the irreducible components Θ i and Θ j ( i ≠ j ) are connected by ˆΘ i,j = deg ( Θ i ⋅ Θ j ) edges. Definition C.7 (Kodaira symbols, See [46, Theorem 6.3]) . Kodaira has introduced the followingsymbols characterizing the type of one-cycles appearing in the study of minimal elliptic surfaces.See Table 4 for a visualization of these fibers.1. Type I : a smooth curve of genus 1.2. Type I : an irreducible nodal rational curve.3. Type II: an irreducible cuspidal rational curve.4. Type I : Θ = Θ + Θ and Θ ⋅ Θ = p + p : two smooth rational curves intersecting transversallyat two distinct points p and p . The dual graph of I is ̃ A .38. Type III: Θ = Θ + Θ and Θ ⋅ Θ = p : two smooth rational curves intersecting at a doublepoint. Its dual graph is ̃ A .6. Type IV: Θ = Θ + Θ + Θ and Θ ⋅ Θ = Θ ⋅ Θ = Θ ⋅ Θ = p : a 3-star composed of smoothrational curves. Its dual graph is ̃ A .7. Type I n ( n ≥ ) : Θ = Θ + ⋯ Θ n with Θ i ⋅ Θ i + = p i i = , ⋯ , n − and Θ n ⋅ Θ = p n . Its dualgraph is the affine Dynkin diagram ̃ A n − .8. Type I ∗ n ( n ≥ ) : Θ = Θ + Θ + +⋯+ n + + Θ n + + Θ n + , with Θ i ⋅ Θ i + = p i ( i = , . . . , n + ) , Θ ⋅ Θ = p , Θ n + ⋅ Θ n + = p n + . The dual graph the affine Dynkin diagram ̃ D + n .9. Type IV ∗ : Θ = Θ + Θ + + + + + Θ with Θ i ⋅ Θ i + = p i ( i = , . . . , ), Θ ⋅ Θ = p , Θ ⋅ Θ = p , Θ ⋅ Θ = p . The dual graph is the affine Dynkin diagram ̃ E .10. Type III ∗ : Θ = Θ + + + + + + + Θ with Θ i ⋅ Θ i + = p i ( i = , . . . , ), Θ ⋅ Θ = p , Θ ⋅ Θ = p , Θ ⋅ Θ = p . The dual graph is the affine Dynkin diagram ̃ E .11. Type II ∗ : Θ = + + + + + + + + Θ , with Θ i ⋅ Θ i + = p i ( i = , . . . , ), Θ ⋅ Θ = p , Θ ⋅ Θ = p , and Θ ⋅ Θ = p . The dual graph the affine Dynkin diagram ̃ E . C.2 Elliptic fibrations, generic versus geometric fibers
Definition C.8 (Elliptic fibrations) . A surjective proper morphism ϕ ∶ Y Ð→ B between twoalgebraic varieties Y and B is called an elliptic fibration if the generic fiber of ϕ is a smoothprojective curve of genus one and ϕ has a rational section. When B is a curve, Y is called an ellipticsurface. When B is a surface, Y is said to be an elliptic threefold. In general, if B has dimension n − , Y is called an elliptic n -fold.The locus of singular fibers of ϕ is called the discriminant locus of ϕ and is denoted ∆ ( ϕ ) orsimply ∆ when the context is clear. If the base B is smooth, the discriminant locus is a divisor [20].The singular fibers of a minimal elliptic surface have been classified by Kodaira and Néron. Thedual graphs of these geometric fibers are affine Dynkin diagrams. We denote these singular fibersby their Kodaira symbols as described in Definition C.7 and presented in Table 4.The language of schemes streamlines many notions in the study of fibrations. We review somebasic definitions. Definition C.9 (Fiber over a point) . Let ϕ ∶ Y Ð→ B be a morphism of schemes. For any p ∈ B ,the fiber over p is denoted Y p and defined using a fibral product as Y p = Y × B Spec κ ( p ) . The first projection Y p Ð→ Y induces an homeomorphism from Y p onto f − ( p ) [50, §3.1 Propo-sition 1.16] . The second projection gives Y p the structure of a scheme over the residue field κ ( p ) . Given three sets ( A , A , and S ) and two maps ϕ ∶ A → B and ϕ ∶ A → B , we define the fibral product A × S A as the subset of A × A composed of couples ( a , a ) such that ϕ ( a ) = ϕ ( a ) . p is not a closed point , the residue field κ ( p ) is not necessarily algebraically closed. Certaincomponents of Y p could be κ ( p ) -irreducible (i.e. irreducible when defined over κ ( p ) ) while theybecome reducible after an appropriate field extension. An irreducible scheme over a field k is saidto be geometrically irreducible when it stays irreducible after any field extension. The most refineddescription of the fiber Y p is always the one corresponding to the algebraic closure κ ( p ) of κ ( p ) .This motivates the following definition. Definition C.10.
The geometric fiber over p is the fiber Y p × κ ( p ) κ ( p ) , the fiber Y p after the basechange induced by the field extension κ ( p ) → κ ( p ) to the algebraic closure of κ ( p ) .By construction, a geometric fiber is always composed of geometrically irreducible components. Definition C.11.
We say that the type of a fiber Y p is geometric if it does not change after a fieldextension. Remark C.12.
To emphasize the difference between the fiber Y p and its geometric fiber, we willrefer to the fiber Y p (defined with respect to the residue field κ ( p ) ) as the arithmetic fiber .For an elliptic n -fold, the Kodaira fibers are also the geometric generic fibers of the irreduciblecomponents of the reduced discriminant locus. While the dual graph of a Kodaira fiber is an affineDynkin diagram of type ̃ A k , ̃ D + k , ̃ E , ̃ E , or ̃ E , the dual graph of the generic (arithmetic) fiberitself can also be a twisted Dynkin diagram of type ̃ B t + k , ̃ C t + k , ̃ G t , or ̃ F t . This is reviewed in Tables2 and 3. These dual graphs are not geometric in the sense that after an appropriate base changethey become ̃ D + n , ̃ A + k or ̃ A + k , and ̃ E respectively. The Kodaira fibers of the following typenever need a field extension: I , II, III, III ∗ , and II ∗ . The remaining Kodaira fibers (IV, I n > , I ∗ n , and IV ∗ ) can come from fibers Y p whose types arenot geometric and require at least a field extension of degree to describe a fiber with a geometrictype. When the fiber Y p has a geometric type, the type of the fiber is said to be split . Otherwise, thetype of Y p is said to be non-split. When that is the case we mark the fiber with an “ns” superscript:IV ns , I ns n , I ∗ ns n , ( n ≥ ) and IV ∗ ns . When a field extension is not needed, the fibers are markedwith an “s” superscript (“split”): IV s , I s n , I ∗ s n , ( n ≥ ) and IV ∗ s . The fiber of type I ∗ can be split,semi-split, or non-split if the Kodaira types require no field extension, a quadratic extension, or acubic extension. The corresponding dual graphs are respectively ̃ D , ̃ B t , and ̃ G t . C.3 Weierstrass models and Deligne’s formulaire
We follow the notation of Deligne [16]. Let L be a line bundle over a quasi-projective variety B .We define the following projective bundle (of lines): π ∶ X = P B [ O B ⊕ L ⊗ ⊕ L ⊗ ] Ð→ B. (C.1)The relative projective coordinates of X over B are denoted [ z ∶ x ∶ y ] , where z , x , and y are definedrespectively by the natural injection of O B , L ⊗ , and L ⊗ into O B ⊕ L ⊗ ⊕ L ⊗ . Hence, z is asection of O X ( ) , x is a section of O X ( ) ⊗ π ∗ L ⊗ , and y is a section of O X ( ) ⊗ π ∗ L ⊗ . For example, if p is the generic point of a subvariety of B . efinition C.13. A Weierstrass model is an elliptic fibration ϕ ∶ Y → B cut out by the zero locusof a section of the line bundle O ( ) ⊗ π ∗ L ⊗ in X .The most general Weierstrass equation is written in the notation of Tate as [16] F = with F = y z + a xyz + a yz − ( x + a x z + a xz + a z ) , (C.2)where a i is a section of π ∗ L ⊗ i . The line bundle L is called the fundamental line bundle of theWeierstrass model ϕ ∶ Y → B and can be defined directly from the elliptic fibration Y as L = R ϕ ∗ O Y . Following Tate and Deligne, we introduce the following quantities [16] ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ b = a + a b = a a + a b = a + a b = a a − a a a + a a + a a − a c = b − b c = − b + b b − b ∆ = − b b − b − b + b b b j = c / ∆ (C.3)These quantities satisfy the following two relations = c − c , b = b b − b . (C.4)The b i ( i = , , , ) and c i ( i = , ) are sections of π ∗ L ⊗ i . The discriminant ∆ is a section of π ∗ L ⊗ . Geometrically, the discriminant ∆ is the locus of points over which the elliptic fiber issingular. The j -invariant characterizes a smooth elliptic curve up to isomorphism. If we completethe square in y in the Weierstrass equation, the equation becomes zy = x + b x z + b xz + b z . (C.5)In addition, if we complete the cube in x gives the short form of the Weierstrass equation, theequation becomes zy = x − c xz − c z . (C.6) C.4 Tate’s algorithm
Let R be a complete discrete valuation ring with valuation v , uniformizing parameter s , and perfectresidue field κ = R /( s ) . We are interested in the case where κ has characteristic zero. We recallthat a discrete valuation ring has only three ideals, the zero ideal, the ring itself, and the principalideal sR . We take the convention in which the ring itself is not a prime ideal. It follows that thescheme Spec ( R ) has only two points: the generic point (defined by the zero ideal) and the closedpoint (defined by the principal ideal sR ). 41et E / R be an elliptic curve over R with Weierstrass equation y + a xy + a y = x + a x + a x + a , a i ∈ R. The generic fiber is a regular elliptic curve. After a resolution of singularities, we have a regularmodel E over R and the special fiber is the fiber over the closed point Spec R /( s ) . Tate’s algorithmdetermines the type of the geometric fiber over the closed point of Spec ( R ) by manipulating thevaluations of the coefficients and the discriminant, and the arithmetic properties of some auxiliarypolynomials. The type of the geometric fiber is denoted by its Kodaira’s symbol (see Definition C.7).The special fiber becomes geometric after a quadratic or a cubic field extension κ ′ / κ . Keeping trackof the field extension used gives a classification of the special fiber as a κ -scheme—this is what we callthe arithmetic fiber. The information on the required field extension needed to have geometricallyirreducible components is already carefully encoded in Tate’s original algorithm, as it is needed tocompute the local index (denoted by c in Tate’s notation). In the language of Néron’s model, thelocal index c is the order of the component group; geometrically, the local index is the number ofreduced components of the special fiber defined over κ . . Following Tate, we use the convenientnotation a i,j = a i s − j . Tate’s algorithm consists of the following eleven steps (see [70], [67, §IV.9], [41], [19], [23]). ForStep 7, we use the more refined description of Papadopoulos [41, Part III, page 134] who also givesin [41, §1, page 122] an exhaustive list of errata of Tate’s original paper [70]. Tate’s algorithm isdiscussed in F-theory in [8, 45]. Subtleties in Step 6 and the distinction between two G -modelsdepending on [ κ ′ ∶ κ ] are explained in [27]. We follow the presentation of [23]:Step 1. v ( ∆ ) = Ô⇒ I .Step 2. If v ( ∆ ) ≥ , change coordinates so that v ( a ) ≥ , v ( a ) ≥ , and v ( a ) ≥ .If v ( b ) = , the type is I v ( ∆ ) . To have a fiber with geometric irreducible components, it isenough to work in the splitting field κ ′ of the following polynomial of κ [ T ] : T + a T − a . The discriminant of this quadric is b . If b is a square in κ , then κ ′ = κ , otherwise κ ′ ≠ κ :(a) κ ′ = κ Ô⇒ I s n (b) κ ′ ≠ κ Ô⇒ I ns n Step 3. v ( b ) ≥ , v ( a ) ≥ , v ( a ) ≥ , and v ( a ) = Ô⇒ II.Step 4. v ( b ) ≥ , v ( a ) ≥ , v ( a ) = , and v ( a ) ≥ Ô⇒ III.Step 5. v ( b ) ≥ , v ( a ) ≥ , v ( a ) ≥ , v ( a ) ≥ , and v ( b ) = Ô⇒ IV.The fiber has geometric irreducible components over the splitting field κ ′ of the polynomial T + a , T − a , Its discriminant is b , . If b , is a square in κ , then κ ′ = κ otherwise κ ′ ≠ κ .(a) κ ′ = κ Ô⇒ IV s (b) κ ′ ≠ κ Ô⇒ IV ns v ( b ) ≥ , v ( a ) ≥ , v ( a ) ≥ , v ( a ) ≥ , v ( b ) ≥ , v ( b ) ≥ . Then make a change ofcoordinates such that v ( a ) ≥ , v ( a ) ≥ , v ( a ) ≥ , v ( a ) ≥ , and v ( a ) ≥ . Let P ( T ) = T + a , T + a , T + a , If P ( T ) is a separable polynomial in κ , that is if P ( T ) has three distinct roots in a fieldextension of κ , then the type is I ∗ . The geometric fiber is defined over the splitting field κ ′ of P ( T ) in κ . The type of the special fiber before to go to the splitting field depends on thedegree of the field extension κ ′ → κ :• [ κ ′ ∶ κ ] = or Ô⇒ I ∗ ns with dual graph ̃ G t .• [ κ ′ ∶ κ ] = Ô⇒ I ∗ ss with dual graph ̃ B t .• [ κ ′ ∶ κ ] = Ô⇒ I ∗ s with dual graph ̃ D .where “ns”, “ss”, and “s” stand respectively for “non-split”, “semi-split”, and “split”. In thenotation of Liu, these fibers are respectively I ∗ , , I ∗ , , and I ∗ . The Galois group is thesymmetric group S , the cyclic group Z / Z , the cyclic group Z / Z or the identity when thedegree is respectively , , , and .Step 7. If P ( T ) has a double root, then the type is I ∗ n with n ≥ . Make a change of coordinates suchthat the double root is at the origin. Then v ( a ) ≥ , v ( a ) = , v ( a ) ≥ , v ( a ) ≥ , , v ( a ) ≥ , and v ( ∆ ) = n + ( n ≥ ) . We now assume that, except for their valuations, theWeierstrass coefficients are generic. We then distinguish between even and odd values of n .(a) If n = ℓ − ( ℓ ≥ ), then v ( a ) ≥ , v ( a ) = , v ( a ) ≥ ℓ , v ( a ) ≥ ℓ + , v ( a ) ≥ ℓ , v ( b ) = ℓ , v ( b ) = ℓ + , and T + a ,ℓ T − a , ℓ has two distinct roots in its splitting field κ ′ . If the two roots are rational ( [ κ ′ ∶ κ ] = )then we have I ∗ s ℓ − with dual graph ̃ D ℓ + , otherwise ( [ κ ′ ∶ κ ] = ) we have the fiber typeI ∗ ns ℓ − with dual graph ̃ B t ℓ .(b) If n = ℓ − ( ℓ ≥ ) then, v ( a ) ≥ , v ( a ) = , v ( a ) ≥ ℓ + , v ( a ) ≥ ℓ + , v ( a ) ≥ ℓ + ,and v ( b ) = ℓ + . The polynomial a , T + a ,ℓ + T − a , ℓ + has two distinct roots in its splitting field. If the two roots are rational then we have I ∗ s ℓ − with dual graph ̃ D ℓ + , otherwise I ∗ ns ℓ − with dual graph ̃ B t ℓ + .Step 8. If P ( T ) has a triple root, change coordinates such that the triple root is zero. 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