Lagrangian 4-planes in holomorphic symplectic varieties of K3^[4] type
aa r X i v : . [ m a t h . AG ] A ug LAGRANGIAN -PLANES IN HOLOMORPHIC SYMPLECTIC VARIETIES OF K [4] -TYPE BENJAMIN BAKKER AND ANDREI JORZA
Abstract.
We classify the cohomology classes of Lagrangian 4-planes P in a smooth manifold X deforma-tion equivalent to a Hilbert scheme of 4 points on a K K S is generated by nonegative classes C , for which ( C, C ) ≥
0, andnodal classes C , for which ( C, C ) = −
2; Hassett and Tschinkel conjecture that the cone of effective curves ona holomorphic symplectic variety X is similarly controlled by “nodal” classes C such that ( C, C ) = − γ , for( · , · ) now the Beauville-Bogomolov form, where γ classifies the geometry of the extremal contraction associ-ated to C . In particular, they conjecture that for X deformation equivalent to a Hilbert scheme of n pointson a K C = ℓ of a line in a smooth Lagrangian n -plane P n must satisfy ( ℓ, ℓ ) = − n +32 .We prove the conjecture for n = 4 by computing the ring of monodromy invariants on X , and showing thereis a unique monodromy orbit of Lagrangian 4-planes. Introduction
Let X be an irreducible holomorphic symplectic variety; thus, X is a smooth projective simply-connectedvariety whose space H (Ω X ) of global two-forms is generated by a nowhere degenerate form ω . H ( X, Z )carries a deformation-invariant nondegenerate primitive integral form ( · , · ) called the Beauville-Bogomolovform [Bea83]. For X = S a K · , · ) is the intersection form, while for X = S [ n ] a Hilbert scheme of n > S we have the orthogonal decomposition [Bea83, § H ( S [ n ] , Z ) ( · , · ) ∼ = H ( S, Z ) ⊕ ⊥ Z δ (1)where the form on H ( S, Z ) is the intersection form, 2 δ is the divisor of non-reduced subschemes, and( δ, δ ) = 2 − n . The embedding of H ( S, Z ) is achieved via the canonical isomorphism H ( S, Z ) ∼ = H (Sym n S, Z )and pullback along the contraction σ : S [ n ] → Sym n S . The inverse of ( · , · ) defines a Q -valued form on H ( X, Z ) which we will also denote ( · , · ); by Poincar´e duality, we obtain a decomposition dual to (1). Forexample, the class δ ∨ ∈ H ( X, Z ) Poincar´e dual to the exceptional divisor δ has square ( δ ∨ , δ ∨ ) = − n .The form induces an embedding H ( X, Z ) ⊂ H ( X, Z ) under which the two forms match up, and since thedeterminant of ( · , · ) on H ( X, Z ) is 2 − n , we can write any ℓ ∈ H ( X, Z ) as ℓ = λ n − for some λ ∈ H ( X, Z ).We will refer to the smallest multiple of ℓ that is in H ( X, Z ) as the Beauville-Bogomolov dual ρ of ℓ .1.1. Cones of effective curves.
Much of the geometry of a K S is encoded in its nodal classes,the indecomposable effective curve classes C for which ( C, C ) = −
2. Suppose S has an ample divisor H ;let N ( S, Z ) ⊂ H ( S, Z ) be the group of curve classes modulo homological equivalence, and NE ( S ) ⊂ N ( S, R ) = N ( S, Z ) ⊗ R the cone of effective curves. Then it is well-know that [LP80, Lemma 1.6]NE ( S ) = h C ∈ N ( S, Z ) | H · C > C · C ≥ − i (2)By Kleiman’s criterion there is a dual statement for the ample cone; here by h· · · i we mean “the conegenerated by · · · ”.Hassett and Tschinkel [HT10b] conjectured that the cone of effective curves in a holomorphic symplecticvariety X is similarly determined intersection theoretically by the Beauville-Bogomolov form. The originalform of this conjecture was: Date : October 15, 2018. isproven Conjecture 1. ( [HT10b, Thesis 1.1] ) Let X be an irreducible holomorphic symplectic varietywith polarization H . Then there is a positive rational constant c X dependent only on the deformation classof X such that NE ( X ) = h C ∈ N ( X, Z ) | H · C >
C, C ) ≥ − c X i Further, if X contains a smoothly embedded Lagrangian n -plane P n ⊂ X , and ℓ ∈ NE ( X ) is the class ofthe line in P n , then the bound is realized: ( ℓ, ℓ ) = − c X Remark . As stated the first part of this conjecture is false. A counterexample was originally constructedby Markman [Mar] for X deformation equivalent to a Hilbert scheme of 5 points on a K X of dimension < X deformation equivalent to a Hilbert scheme of 2 points on a K − , − , and − —and their extremal rays correspond to the 2 types of extremal contractions:(i) Divisorial extremal contractions. In this case, the exceptional divisor E is contracted to a K T . The generic fiber over T is either an A or A configuration of rational curves [HT09, Theorem21], and if C is the class of the generic fiber of the normalization, then either ( C, C ) = − − / f contracts a Lagrangian P to an isolated singularity,and the class of a line ℓ satisfies ( ℓ, ℓ ) = − / Lagrangian n -planes. Generalizing slightly, let X be an irreducible holomorphic symplectic mani-fold —that is, a simply-connected K¨ahler manifold with H (Ω X ) ∼ = C generated by a nowhere degenerate2-form. There are only two infinite families of deformation classes of irreducible holomorphic symplecticmanifolds known: Hilbert schemes of points on K X to be of K [ n ] -type if it is deformationequivalent to a Hilbert scheme of n points on a K Conjecture 2. ( [HT10b, Conjecture 1.2] ) Let X be of K [ n ] -type, P n ⊂ X a smoothly embedded Lagrangian n -plane, and ℓ ∈ H ( X, Z ) the class of the line in P n . Then ( ℓ, ℓ ) = − n + 32Further, we expect this to still be the minimal Beauville-Bogomolov square of a reduced irreducible curveclass. The conjecture has been verified for n = 2 in [HT09] and for n = 3 in [HHT]. Remark . There is a similar conjecture for the class of a line ℓ in a smoothly embedded Lagrangian n -plane P n ⊂ X for X deformation equivalent to a 2 n -dimensional generalized Kummer variety K n A of anabelian surface A . In this case, we expect ( ℓ, ℓ ) = − n + 12This conjecture has been verified for n = 2 in [HT10a].Our main result is a proof of Conjecture 2 in the n = 4 case; furthermore, we completely classify the classof the Lagrangian 4-plane: Theorem 1.5 (see Theorem 4.4) . Let X be of K [4] -type, P ⊂ X be a smoothly embedded Lagrangian4-plane, ℓ ∈ H ( X, Z ) the class of a line in P , and ρ = 2 ℓ ∈ H ( X, Q ) . Then ρ is integral, and [ P ] = 1337920 (cid:0) ρ + 1760 ρ c ( X ) − θ + 4928 θ c ( X ) − ( X ) (cid:1) Further, we must have ( ℓ, ℓ ) = − . ere θ is the image of the dual to the Beauville-Bogomolov form, thought of as an element of Sym H ( X, Q ) ∗ ∼ =Sym H ( X, Q ), under the cup product map Sym H ( X, Q ) → H ( X, Q ). Likewise in the n = 3 case theclass of the Lagrangian 3-plane is completely determined by ℓ , cf. [HHT, Theorem 1.1]. Our theorem providesevidence that Conjecture 2 is true in general, and conjecturally determines the minimal Beauville-Bogomolovsquare of indecomposable nodal classes on eightfolds deformation equivalent to Hilbert schemes of points on K Monodromy.
We prove our result by using the representation theory of the monodromy group of X to relate the intersection theory of X to that of a Hilbert scheme of 4 points on a K X .Recall that a monodromy operator is the parallel translation operator on H ∗ ( X, Z ) associated to a smoothfamily of deformations of X ; the monodromy group Mon( X ) is the subgroup of GL( H ∗ ( X, Z )) generated byall monodromy operators. Let Mon ( X ) ⊂ GL( H ( X, Z )) be the quotient acting nontrivially on degree 2cohomology, and Mon( X ) ⊂ GL( H ∗ ( X, C )) (respectively Mon ( X ) ⊂ GL( H ( X, C ))) the Zariski closure ofMon( X ) (respectively Mon ( X )). By the deformation invariance of the Beauville-Bogomolov form, Mon ( X )is actually contained in O( H ( X, Z )), the orthogonal group of H ( X, Z ) with respect to ( · , · ). A priori ,the full Lie group G X = SO( H ( X, C )) only acts on H ( X, C ), but in fact for X of K [ n ] -type, the fullcohomology ring H ∗ ( X, C ) carries a representation of G X = SO( H ( X, C )) compatible with cup product([HHT, Proposition 4.1]). The basic reason for this is two-fold, both results of Markman:(a) the quotient Mon( X ) → Mon ( X ) has finite kernel [Mar08, § . G X is a connected component of Mon ( X ) [Mar08, § . X ) on H ∗ ( X, C ) extends to one of Mon( X ). By the above the connected com-ponent of the universal covers of Mon( X ) , Mon ( X ) and G X are all identified, so the universal cover of G X acts on all of H ∗ ( X, C ); the representation descends to G X because of the vanishing of odd cohomology.The action respects the Hodge structure, so we may consider the ring of Hodge classes: I ∗ ( X ) = H ∗ ( X, Q ) ∩ H ∗ ( X, C ) G X Of course, I ∗ ( X ) contains the Chern classes of the tangent bundle of X and the Beauville-Bogomolov class θ ∈ H ( X, Q ), but there can be many other Hodge classes. Markman [Mar11] constructs another series ofHodge classes k i ∈ I i ( X ), i ≥
2, as characteristic classes of monodromy-invariant twisted sheaves.Given λ ∈ H ( X, Q ), let G λ ⊂ G X be the stabilizer of λ . Define I ∗ λ ( X ) = H ∗ ( X, Q ) ∩ H ∗ ( X, C ) G λ to be the ring of cohomology classes invariant under the monodromy group preserving λ . For example, givena Lagrangian n -plane P n ⊂ X , the deformations of X that deform P n are precisely those in H , ( X ) ∩ ρ ⊥ ,where ρ is the Beauville-Bogomolov dual of the class of the line in P n , and the orthogonal is taken withrespect to the Beauville-Bogomolov form [Ran95, Voi92]. Thus, the class [ P n ] ∈ H n ( X, Z ) must lie in thesubring I ∗ ρ ( X ). G X will act on these cohomology classes, and up to this action we expect there is a uniqueLagrangian n -plane in general. For n = 4, this is a consequence of our result since G X acts transitively onrays in H ( X, C ): Corollary 1.7.
For X of K [4] -type, there is a unique G X orbit of smooth Lagrangian -plane classes [ P ] ∈ H ( X, C ) . Method of Proof and Outline.
We prove our result by first completely determining I ∗ λ ( X ) for X = S [4] a Hilbert scheme of 4 points on a K S and λ = δ . This is done in Section 1 using the Nakajimabasis and the results of [LS03] on cup product. The ring I ∗ λ ( X ) in the general case of X of K [4] -type and λ ∈ H ( X, Z ) will be isomorphic since G X acts transitively on rays in H ( X, Z ). In Section 2 we constructan explicit isomorphism by finding a monodromy invariant basis for I ∗ λ ( X ), from which we are able to derivethe intersection form on I λ ( X ). In Section 3 we take λ proportional to the Beauville-Bogomolov dual of theclass of a line in a smooth Lagrangian 4-plane P ⊂ X and produce a diophantine equation in the coefficientsof the class [ P ] with respect to the basis from Section 2. In Section 4, we show the only solution to the iophantine equation is the conjectural one. For completeness we include an appendix summarizing ourlocalization computations to calculate the Fujiki constants in Section 2. Acknowledgements.
We are grateful to Y. Tschinkel for suggesting the problem, and for many insights.We would also like to thank B. Hassett and M. Thaddeus for useful conversations, and M. Stoll for explain-ing to us how to compute integral points on elliptic curves in Magma. Finally, we thank the referee foruseful comments. The first author was supported in part by NSF Fellowship DMS-1103982. This projectwas completed while the second author was a postdoc at the California Institute of Technology. Some com-putations were performed on William Stein’s server geom.math.washington.edu , supported by NSF grantDMS-0821725. 2.
Structure of the ring of monodromy invariants
The Lehn-Sorger formalism.
We briefly summarize the work of Lehn and Sorger in [LS03] on thecohomology ring of a Hilbert scheme of points on a K A , they constructa Frobenius algebra A [ n ] such that when A = H ∗ ( S, Q ) for S a K A [ n ] is canonically H ∗ ( S [ n ] , Q ).The algebra A = H ∗ ( S, Q ) comes equipped with a form T = − R S : A → Q and a multiplication m : A ⊗ A → A (given by cup-product) such that the pairing ( x, y ) = T ( xy ) is nondegenerate. Thereis also a comultiplication ∆ : A → A ⊗ A adjoint to m with respect to the form T ⊗ T on A ⊗ A . Inthis case ∆ is the push-forward along the diagonal. Writing 1 ∈ H ( S, Z ) for the unit, [pt] ∈ H ( S, Z )for the point class, e , . . . , e as a basis for H ( S, Z ), and e ∨ , . . . , e ∨ for the dual basis with respect to theintersection form, a simple computation using adjointness shows that ∆(1) = − P j e j ⊗ e ∨ j − [pt] ⊗ − ⊗ [pt],∆( e j ) = − e j ⊗ [pt] − [pt] ⊗ e j , ∆( e ∨ j ) = − e ∨ j ⊗ [pt] − [pt] ⊗ e ∨ j and ∆([pt]) = − [pt] ⊗ [pt]. Thus e = − n -fold multiplication m [ n ] : A ⊗ n → A and its adjoint ∆[ n ] : A → A ⊗ n . Note that m [1] = ∆[1] = id, m [2] = m , and ∆[2] = ∆. Lemma 2.2.
Using the previous formulae one obtains: ∆[3](1) = X j X ( e j ) a ⊗ ( e ∨ j ) b ⊗ [pt] c + X [pt] a ⊗ [pt] b ⊗ c ∆[3]( e j ) = X [pt] a ⊗ [pt] b ⊗ ( e j ) c ∆[3]( e ∨ j ) = X [pt] a ⊗ [pt] b ⊗ ( e ∨ j ) c ∆[3]([pt]) = [pt] ⊗ [pt] ⊗ [pt]By [pt] a ⊗ [pt] b ⊗ c ∈ A ⊗ we mean [pt] inserted in the a th and b th tensor factors, and 1 inserted in the c th factor. All unspecified sums in Lemma 2.2 are over bijections { , , } ∼ = −→ { a, b, c } . Proof.
This follows from the relation m [ n ] = m [2] ◦ ( m [ n − ⊗ id) for n ≥ n ] =(∆[ n − ⊗ id) ◦ ∆[2]. (cid:3) Let [ n ] = { k ∈ N | k ≤ n } . Define the tensor product of A indexed by a finite set I of cardinality n as A I := M ϕ :[ n ] ∼ = −→ I A ϕ (1) ⊗ · · · ⊗ A ϕ ( n ) /S n where S n acts by permuting the tensor factors in each summand in the obvious way. A I is a Frobeniusalgebra with multiplication m I and form T I .Note that for (finite) sets U, V and a bijection U → V there is a canonical isomorphism A U → A V , so wecan always choose a bijection of I with some [ k ] to reduce to the usual notion of finite self tensor products.In general, for any surjection ϕ : U → V , there is an obvious ring homomorphism ϕ ∗ : A U → A V using the ring structure to combine factors indexed by elements of U in the same fiber of ϕ . There is anadjoint map ϕ ∗ : A V → A U ith the important relation ϕ ∗ ( a · ϕ ∗ ( b )) = ϕ ∗ ( a ) · b which follows directly from the adjointness.For any subgroup G ⊂ S n , we can consider the left coset space G \ [ n ], and form A G \ [ n ] . In particular, for σ ∈ S n and G = h σ i the group generated by σ , we denote A σ = A G \ [ n ] . Let A { S n } = M σ ∈ S n A σ · σ A pure tensor element of A σ is specified by attaching an element α i ∈ A to each orbit i ∈ I = h σ i\ [ n ]. Forexample, for a function ν : I → Z ≥ , e ν = ⊗ i ∈ I e ν ( i ) ∈ A σ There is a natural product structure on A { S n } . For any inclusion of subgroups H ⊂ K of S n there is asurjection H \ [ n ] → K \ [ n ] and therefore maps f H,K : A H \ [ n ] → A K \ [ n ] f K,H : A K \ [ n ] → A H \ [ n ] The product is then A σ ⊗ A τ −−−−→ A στ a ⊗ b −−−−→ f h σ,τ i , h στ i (cid:0) f h σ i , h σ,τ i ( a ) · f h τ i , h σ,τ i ( b ) · e g ( σ,τ ) (cid:1) (3)where h σ, τ i is the subgroup of S n generated by σ, τ , and the graph defect g ( σ, τ ) : h σ, τ i\ [ n ] → Z ≥ is g ( σ, τ )( B ) = 12 ( | B | + 2 − |h σ i\ B | − |h τ i\ B | − |h στ i\ B | ) S n acts naturally on A { S n } . For any τ ∈ A { S n } , there is for any σ ∈ S n a bijection τ : h σ i\ [ n ] →h τ στ − i\ [ n ]. τ then acts on A { S n } via τ ∗ : A σ · σ → A τστ − · τ στ − on each factor. Define A [ n ] = A { S n } S n Note that for any partition µ = (1 µ , µ , · · · ) of n , there is a piece A [ n ] µ = M σ ∈ C µ A σ · σ S n ∼ = O i Sym µ i A (4)where C µ ⊂ S n is the conjugacy class of permutations σ of cycle type µ .If A is a graded Frobenius algebra, then A [ n ] is naturally graded. A σ is graded as a tensor product ofgraded vector spaces, and we take A σ · σ ∼ = A σ [ − | σ | ]where if the cycle type of σ is µ , | σ | = P i ( i − µ i . In particular, the m th graded piece of (4) is( A [ n ] µ ) m ∼ = M ( w,µ ) | ( w,µ ) | = m O i Sym µ i A w i (5)where the sum is taken over weighted permutations ( w, µ )— i.e. a partition µ and a weight w i associated toeach part—with m = | ( w, µ ) | = X i ( i − µ i + w i We then have
Theorem 2.3. ( [LS03, Theorem 1.1] ) For S a K surface, there is a natural isomorphism of graded Frobeniusalgebras ( H ∗ ( S, Q )[2]) [ n ] ∼ = H ∗ ( S [ n ] , Q )[2 n ]The grading shift on both sides is such that the 0th graded piece is middle cohomology. emark . It will be important in the next section to note that under the isomorphism of Theorem 2.3, n ![pt] ⊗ · · · ⊗ [pt] n · (id) [pt] S [ n ] (6)2.5. Monodromy invariants.
Let S be a K G S = SO( H ( S, C )) the special orthogonalgroup of the intersection form ( · , · ) on S . H ∗ ( S, C ) is naturally a representation of G S , acting via thestandard representation on H ( S, C ) and the trivial representations on H ( S, C ) and H ( S, C ).Recall (see for example [FH91]) that positive weights of the algebra SO C ( k ) of rank r ( k = 2 r or 2 k + 1)are r -tuples λ = ( λ , . . . , λ r ) with the λ i either all integral or all half-integral, and either λ ≥ λ ≥ · · · ≥ λ r − ≥ | λ r | ≥ , k = 2 rλ ≥ λ ≥ · · · ≥ λ r − ≥ λ r ≥ , k = 2 r + 1Let the representation of SO C of highest weight λ be denoted V ( λ ). Thus, = V (0 , . . . ) is the trivialrepresentation, and V = V (1 , , . . . ) the standard. Sym k V is not irreducible, since the form yields aninvariant θ ∈ Sym V , but V ( k, , . . . ) = Sym k V /
Sym k − V . In the sequel, we will only indicate the nonzeroweights, e.g. V = V (1).If a Frobenius algebra A carries a representation of a group G , A [ n ] naturally carries a representation of G that can easily be read off of (5). Thus, Proposition 2.6.
As a representation of G S , we have H ( S [4] , C ) ∼ = S ⊕ V S (1) H ( S [4] , C ) ∼ = S ⊕ V S (1) ⊕ V S (2) H ( S [4] , C ) ∼ = S ⊕ V S (1) ⊕ V S (1 , ⊕ V S (2) ⊕ V S (3) H ( S [4] , C ) ∼ = S ⊕ V S (1) ⊕ V S (1 , ⊕ V S (2) ⊕ V S (2 , ⊕ V S (3) ⊕ V S (4)Poincar´e duality is compatible with the G S action, so the above determines all cohomology groups.Note that the invariant class in H ( S [ n ] , C ) is exactly δ . The decomposition (1) identifies the action of G S on H ∗ ( S [ n ] , C ) with that of G δ ⊂ G S [ n ] , the stabilizer of δ . In other words, deformations of S [ n ] orthogonalto the exceptional divisor δ remain Hilbert schemes of points of a K S .Recall that SO C ( k ) has universal branching rules. For SO C ( k − ⊂ SO C ( k ) the stabilizer of a nonisotropicvector v ∈ V , ( v, v ) = 0, we have Res SO C ( k )SO C ( k − V ( λ ) = M λ ′ V ( λ ′ )where the sum is taken over all weights λ ′ with λ ≥ λ ′ ≥ λ ≥ λ ′ ≥ · · · ≥ λ r ≥ | λ ′ r | ≥ X of K [ n ] -type, we can therefore deduce the structure of H ∗ ( X, C ) as a G X representation from thestructure of H ∗ ( S [ n ] , C ) as a G S representation: Corollary 2.7.
For X of K [4] -type, H ( X, C ) ∼ = V X (1) H ( X, C ) ∼ = X ⊕ V X (1) ⊕ V X (2) H ( X, C ) ∼ = X ⊕ V X (1) ⊕ V X (1 , ⊕ V X (2) ⊕ V X (3) H ( X, C ) ∼ = X ⊕ V X (1) ⊕ V X (2) ⊕ V X (2 , ⊕ V X (4)Again, Poincar´e duality determines the representations of the other cohomology groups.2.8. A basis for I ∗ δ ( S [4] ) . For a partition µ = (1 µ , µ , . . . ) of n , the number of parts of µ is ℓ ( µ ) = P µ i . Bya labelled partition µ we will mean a partition µ and an ordered list of ℓ ( µ ) cohomology classes α ∈ H ∗ ( S, Q ).For example, ( { } , { , } ) is a labelled partition of 4, subordinate to the partition µ = (1 , he unit class to each part of µ . Such a labelled partition µ determines an element of the Lehn-Sorger algebraof H ∗ ( S, Q )[2] by summing over all group elements σ ∈ S n with cycle type µ , for example I ( { } , { , } ) = X (12) ⊗ ⊗ (12)= 1 ⊗ ⊗ (12) + 1 ⊗ ⊗ (13) + 1 ⊗ ⊗ (14)+ 1 ⊗ ⊗ (23) + 1 ⊗ ⊗ (24) + 1 ⊗ ⊗ (34)We can generate homogeneous classes of H ∗ ( S [ n ] , Q ) invariant under G S from partitions of n labelled bycohomology classes { , e, e ∨ , [pt] } , where every time we have a label e , there must be a paired e ∨ label,corresponding to inserting e j and e ∨ j in the corresponding tensor factors and summing over j . For example, I δ ( S [4] ) is spanned by δ = I ( { } , { , } ). Generating sets for I kδ ( S [4] ) for k = 2 , , I δ ( S [4] ) I δ ( S [4] ) I δ ( S [4] ) W = I ( { } , { } ) P = I ( { } ) A = I ( { e } , { e ∨ } ) X = I ( { , } ) Q = I ( { [pt] } , { , } ) B = I ( { } , { [pt] } ) Y = I ( { , , , [pt] } ) R = I ( { } , { , [pt] } ) C = I ( { [pt] } , { } ) Z = I ( { , , e, e ∨ } ) S = I ( { e ∨ } , { e, } ) D = I ( { , [pt] } ) T = I ( { } , { e, e ∨ } ) E = I ( { e, e ∨ } ) F = I ( { , , [pt] , [pt] } ) G = I ( { , e, e ∨ , [pt] } ) H = I ( { e, e, e ∨ , e ∨ } )These classes are all clearly independent, and therefore by the computation of the dimensions of I ∗ δ ( S [4] ) inthe previous section they are bases.2.9. Cup product on I ∗ δ ( S [4] ) . Using (3) we compute the multiplicative structure of I ∗ δ ( S [4] ) in the abovebasis. These computations are straightforward; for example, δ = X (12) ⊗ ⊗ (12) = X (12) (cid:16) ∆(1) , ⊗ ⊗ (id) + 1 , , ⊗ (132)+ 1 , , ⊗ (142) + 1 , , ⊗ (123) + 1 , , ⊗ (124) + 1 ⊗ (12)(34) (cid:17) = − X [pt] ⊗ ⊗ ⊗ (id) − X (12) X j ( e j ) ⊗ ( e ∨ j ) ⊗ ⊗ (id)+ 3 X (123) ⊗ (123) + 2 X (12)(34) ⊗ (12)(34)= − Y − Z + 3 W + 2 X The multiplication table for degree 4 elements is:
W X Y ZW − A − B − C − D − E +4 F +2 G − A − B − C B + 3 C A + 66 CX − D − E + 2 F + G + H D D + 4 EY F GZ F + 2 G + 2 H In particular, note that: δ = ( δ ) = − A − B − C − D − E + 84 F + 30 G + 6 H (7)The multiplication table for A, B, C, D, E, F, G, H is much simpler, B C D E F G HA B C D E F G H where we have identified top cohomology H ( S [4] , Q ) ∼ = Q as usual via the point class [pt] S [4] = 24[pt] ⊗ [pt] ⊗ [pt] ⊗ [pt] (id) from (6). As a consistency check, from Corollary 3.3 we have δ = 105( δ, δ ) = 136080and indeed, from (7), δ = ( − A − B − C − D − E + 84 F + 30 G + 6 H ) = 136080. Note that theremaining classes and products (of cohomological degree divisible by 4, which is all we need) are determinedby Poincar´e duality.2.10. The Beauville-Bogomolov form.
From (1), we can explicitly write down θ in the W, X, Y, Z basis: θ = X j X ( e j ) ⊗ ⊗ ⊗ (id) ! · X ( e ∨ j ) ⊗ ⊗ ⊗ (id) ! − δ = − W − X + 452 Y + 136 Z (8)By direct compoutation, using the results of the previous section, Lemma 2.11. θ = 450225 δ θ = − − δ θ = 84564 = 2349 · ( − δ θ = − · ( − δ = 136080 = 105 · ( − Hodge classes on X Let X be of K [4] -type and λ ∈ H ( X, Q ). The rings I ∗ ( X ) and I ∗ λ ( X ) are isomorphic to the rings I ∗ ( S [4] ) and I ∗ δ ( S [4] ) since the action of G X is transitive on rays, but to construct an explicit isomorphism,we must find a geometric basis. To do this, we need to understand the products of Hodge classes.3.1. Computation of the Fujiki constants for S [4] . Let X be smooth variety of dimension n , and µ apartition of a nonegative integer | µ | (we allow the empty partition of 0). To each µ we can associate a Chernmonomial c µ ( X ) = Q ki =1 c µ k k ( X ). Given a formal power series ϕ ( x ) ∈ Q [[ x ]], define the associated genus ϕ ( X ) = Y i ϕ ( x i ) ∈ H ∗ ( X, Q )where the x i are the Chern roots of the tangent bundle T X . Taking the universal formal power seriesΦ( x ) = 1 + a x + a x + · · · ∈ Q [ a , a , . . . ][[ x ]]we define the universal genus Φ( X ) of any smooth variety as an element of H ∗ ( X, Q )[ a , a . . . ]. Φ( X ) is auniversal formal power series in the Chern classes c , c , . . . with coefficients polynomials in a , a , . . . . Inparticular, taking a = 1 and a i = 0 for i >
1, we get the total Chern class. We will only need the universalgenus for vanishing odd Chern classes; the reader may find the expansion of Φ in this case up to degree 16in the appendix. et S be a smooth surface, ϕ ( x ) ∈ Q [[ x ]] a formal power series in x . Recall that O [ n ] is the push-forwardof the structure sheaf of the universal subscheme Z ⊂ S × S [ n ] to S [ n ] , and that det O [ n ] = − δ . A result of[EGL01, Theorem 4.2] implies that there are universal formal power series A ( z ) , B ( z ) in z such that X n ≥ z n Z S [ n ] exp(det O [ n ] ) ϕ ( S [ n ] ) = A ( z ) c ( S ) B ( z ) c ( S ) for any smooth surface S . Let F S ( z ) = X n ≥ z n Z S [ n ] exp(det O [ n ] )Φ( S [ n ] ) ∈ Q [ a , a , . . . ][[ z ]]and let A ( z ) , B ( z ) ∈ Q [ a , a , . . . ][[ z ]] be the universal power series associated to Φ. F P ( z ) = A ( z ) B ( z ) and F P × P ( z ) = A ( z ) B ( z ) can be easily computed by routine equivariant localization and therefore onecan compute A ( z ) , B ( z ); see the appendix for a brief summary of the computation. Since P × P , P generatethe cobordism ring, this determines F S ( z ) for a K S , and in particular we can compute all products Z S [ n ] δ k c µ ( S [ n ] ) (9)By the following result of Fujiki, (9) determines all products of the form Z X f k c µ ( X )for arbitrary f ∈ H ( X, Q ): Theorem 3.2. [Fuj87]
For X an irreducible holomorphic symplectic variety of dimension n and µ an evenpartition of an integer | µ | , there are rational constants γ X ( µ ) such that, for any class f ∈ H ( X, Z ) , Z X f k c µ ( X ) = γ X ( µ ) · ( f, f ) k , | mboxf or k = 2 n − | µ | Moreover, the constant γ X ( µ ) is a deformation invariant. Of course, if | µ | > dim X , we have γ X ( µ ) = 0. Also, because X is holomorphic symplectic, all odd Chernclasses c i ( X ) vanish, so we require µ to be an even partition. We collect here the Fujiki constants γ ( µ ) for n = 4 for reference: Corollary 3.3.
For X of K [4] -type, we have γ X (2 ) = 1992240 γ X (2 ) = 59640 γ X (2 ) = 4932 γ X (2 ) = 630 γ X ( ∅ ) = 105 γ X (2 ) = 813240 γ X (2 ) = 24360 γ X (4 ) = 2016 γ X (2 ) = 182340 γ (6 ) = 5460 γ X (8 ) = 25650 γ X (4 ) = 332730 Proof.
This follows from the deformation invariance and the degree 4 part of F S ( z ) = B ( z ) for S a K δ, δ ) = − (cid:3) Remark . The first column of numbers are the Chern numbers of X , and were computed in [EGL01]; γ X ( ∅ ) is the ordinary Fujiki constant. The authors are unaware of a computation of the middle threecolumns in the literature.3.5. Generalized Fujiki constants.
Let X be of K [ n ] -type. In general, for η a Hodge class, an integralof the form R X f k η must be compatible with the G X action, and therefore will be a rational multiple of( f, f ) k . For η a product of a power of θ and a Chern monomial, these ratios are determined by the Fujikiconstants of the previous section.Define an augmented partition ( ℓ, µ ) to be a partition µ of a nonnegative integer | µ | and a nonegativeinteger ℓ . Set | ( ℓ, µ ) | = 2 ℓ + | µ | roposition 3.6. For X of K [ n ] -type, n > , and ( ℓ, µ ) an augmented even partition, there is a rationalconstant γ X ( ℓ, µ ) such that for any f ∈ H ( X, Z ) , Z X f k θ ℓ c µ ( X ) = γ X ( ℓ, µ ) · ( f, f ) k , for k = 2 n − ℓ − | µ | Furthermore, there are rational constants α ( k, ℓ ) independent of X such that γ X ( ℓ, µ ) = α ( k, ℓ ) γ X ( µ ) , for k = 2 n − ℓ − | µ | Again, γ X ( k, ℓ, µ ) = 0 if | ( ℓ, µ ) | > dim X . Proof.
As mentioned above, the interesting part is the existence of the α . Let x i be an orthonormal basisof H ( X, C ) with respect to the Beauville-Bogomolov form. Note that θ = P i x i . It suffices to consider thecase f = P i x i , which has ( f, f ) = 23. Let p k ( a ) = X i a i x i ! k for a ∈ Q . The p k ( a ) span the space of degree k polynomials in x i , so their symmetrizations p k ( a ) = 123! X σ ∈ S X i a i x σ ( i ) ! k span the space of degree k symmetric functions in x i . We can therefore write f k θ ℓ = X a ( k,ℓ ) λ a ( k,ℓ ) p k +2 ℓ ( a ( k, ℓ ))where the sum is over finitely many a ( k, ℓ ). This expression has no dependence on the dimension of X . Wehave Z X f k θ ℓ c µ ( X ) = 123! X a ( k,ℓ ) λ a ( k,ℓ ) X σ ∈ S Z X X i a ( k, ℓ ) i x σ ( i ) ! k +2 ℓ c µ ( X )= 123! X a ( k,ℓ ) λ a ( k,ℓ ) X σ ∈ S X i a ( k, ℓ ) i x σ ( i ) , X i a ( k, ℓ ) i x σ ( i ) ! k + ℓ γ X ( µ )= X a ( k,ℓ ) λ a ( k,ℓ ) X i a ( k, ℓ ) i ! k + ℓ γ X ( µ )= α ( k, ℓ ) γ X ( µ )( f, f ) k + ℓ where α ( k, ℓ ) = 123 k X a ( k,ℓ ) λ a ( k,ℓ ) X i a ( k, ℓ ) i ! k + ℓ (cid:3) Explicitly, Z X θ c µ ( X ) = X i Z X x i c µ ( X ) = X i ( x i , x i ) γ X ( µ ) = 23 · γ X ( µ )so α (0 ,
1) = 23. Less trivially, Z X θ c µ ( X ) = Z X X i x i ! c µ ( X )= Z X X i We have α (0 , 1) = 23 α (1 , 1) = α (2 , 1) = α (3 , 1) = α (0 , 2) = α (1 , 2) = 45 α (2 , 2) = α (0 , 3) = 1035 α (1 , 3) = α (0 , 4) = Of course, α ( k, 0) = 1 for any k . Proof. α (3 , , α (2 , , α (1 , , α (0 , 4) are all determined by Lemma 2.11, using γ S [4] ( ∅ ) = 105. Because α ( k, ℓ ) is independent of the dimension of X , we can determine the remaining α constants from the compu-tations of [HHT] in the K [3] -type cases, where( θ S [3] ) = 15525 = 1035 · γ S [3] ( ∅ )( δ S [3] ) ( θ S [3] ) = − · ( δ S [3] , δ S [3] ) · γ S [3] ( ∅ )( δ S [3] ) ( θ S [3] ) = 1296 = 275 · ( δ S [3] , δ S [3] ) · γ S [3] ( ∅ )( θ S [3] ) c ( S [3] ) = 20700 = 5753 · γ S [3] (2 )( δ S [3] ) ( θ S [3] )c ( S [3] ) = − · ( δ S [3] , δ S [3] ) · γ S [3] (2 )since γ S [3] ( ∅ ) = 15 , γ S [3] (2 ) = 108 and c ( S [3] ) = θ S [3] . (cid:3) A geometric basis. I ( X ) is 3-dimensional, so we expect there to be a relation among θ , θ c ( X ) , c ( X ) , c ( X ): Lemma 3.9. For X of K [4] -type, θ = 75 θc − c + 115 c (10) Proof. Using the results of the previous section, we know the intersection form restricted to I ( X ) in termsof the basis θ , θ c ( X ) , c ( X ) , c ( X ): · · · · · · · · · · · (11)As expected, the matrix is rank 3. By Poincar´e duality, a generator of the kernel gives the relation. (cid:3) Corollary 3.10. c ( S [4] ) = 3 Z + 33 Y − W Proof. Suppose c ( S [4] ) = wW + xX + yY + zZ for w, x, y, z ∈ Q . Taking the product with θ , δ θ , δ θ, δ yields the equation − − − − − − − − wxyz = − − The matrix has rank 2. Computing generators of the kernel, we can writec ( S [4] ) = (cid:18) − u − v (cid:19) W + (cid:18) u − (cid:19) X + ( v + 42) Y − v Z Similarly, computing c ( S [4] ) and intersecting with θ , δ θ, δ yields 3 equations: 945300 = θ c ( S [4] ) , − θδ c ( S [4] ) and 177552 = δ c ( S [4] ) which have exactly two common solutions: ( u, v ) =( , − , ( , − ). Finally, only one of these solutions, ( u, v ) = ( , − ( S [4] ) , and this gives the desired equation. (cid:3) ecall that I ( X ) is 2-dimensional, whereas I λ ( X ) is 4-dimensional. We already have λ ∈ I λ ( X ). Weneed one more geometrically defined class in I λ ( X ) independent from λ and I ( X ) to get a basis for I λ ( X ): Definition 3.11. Given a class λ ∈ H ( X, Q ) (with ( λ, λ ) = 0 so no power of λ is zero), define α ∈ I λ ( X )by Poincar´e duality to be the unique class (up to a multiple) that intersects trivially with λ and I ( X ). Lemma 3.12. For X = S [4] and λ = δ , we may take α = X − Y + Z which intersects trivially with δ θ, δ c ( S [4] ) , δ θ , δ θ c ( S [4] ) , δ c ( S [4] ) , θ , θ c ( S [4] ) , θ c ( S [4] ) , c ( S [4] ) Further, α θ = 9450 , α θ c ( S [4] ) = 14148 and α c ( S [4] ) = 21168 .Proof. By intersecting with θ and c ( S [4] ) using Corollary 3.3 and Lemma 3.7, we see that θ and θ c ( S [4] )are independent in I ( S [4] ), so it is enough to show that α intersects these two classes to conclude itintersects trivially with each of the four degree 12 Hodge classes at the end of the list. This, along withall the other claimed products, follow from Corollary 3.10, equation (8), and our knowledge of the productstructure. Indeed, α = − G + 30 D + 42 F + 3 H + 6 Eαδ = − B + 162 Cαθ = 88 D + 8 E − C + 3 B − F + 20 G + 4 Hα c ( S [4] ) = − C + 132 D + 6 B + 12 E + 30 G + 6 H − Fθ = − E + 192 H + 2152 G − D − A − B + 1117 F − Cδ = − A − B − C − D − E + 84 F + 30 G + 6 Hθ c ( S [4] ) = − A − C − B − E + 1630 F + 153 G + 13 H − D c ( S [4] ) = 18 H − E − B − D + 218 G − A − C + 2380 F and the pairwise products are easily computed. (cid:3) Because the cup-product structure on H ∗ ( S [4] , Z ) is preserved under deformation, and the monodromygroup acts transitively on rays in H ( S [4] , Q ), we immediately conclude the same for arbitrary λ : Corollary 3.13. For α chosen as in Definition 3.11 with respect to λ ∈ H ( X, Z ) , α intersects trivially with λ θ, λ c ( X ) , λ θ , λ θ c ( X ) , λ c ( X ) , θ , θ c ( X ) , θ c ( X ) , c ( X ) Further, up to a rational square, α θ = 9450 , α θ c ( X ) = 14148 and α c ( X ) = 21168 . Middle cohomology. Putting Lemma 3.7 and Corollaries 3.3, and 3.13 together, we now know thecomplete intersection form on middle cohomology I λ ( X ) with respect to the basis: λ , λ θ, λ c ( X ) , θ , θ c ( X ) , c ( X ) , αθ, α c ( X ) (12)Denoting it by M ( λ ), it is: λ, λ ) λ, λ ) λ, λ ) λ, λ ) λ, λ ) λ, λ ) λ, λ ) λ, λ ) λ, λ ) λ, λ ) 28350( λ, λ ) 44110( λ, λ )630( λ, λ ) λ, λ ) λ, λ ) λ, λ ) 44110( λ, λ ) 59640( λ, λ )2349( λ, λ ) λ, λ ) 28350( λ, λ ) 450225 652050 9453003402( λ, λ ) λ, λ ) 41100( λ, λ ) 652050 945300 13717204932( λ, λ ) λ, λ ) 59640( λ, λ ) 945300 1371720 1992240 9450 1414814148 21168 Note that this matrix is nonsingular if ( λ, λ ) = 0, and therefore (12) is in fact a basis. . Lagrangian n -planes in X Let X be a 2 n dimensional holomorphic symplectic variety, and suppose that P n ⊂ X is a smoothlyembedded Lagrangian n -plane. By a simple calculation, Lemma 4.1. [HHT] Denote by h the hyperplane class on P n . Then in the above setup, c j ( T X | P n ) = ( − j h j (cid:18) n + 1 j (cid:19) Proof. We have 0 → T P n → T X | P n → N P n /X → P n is Lagrangian, N P n /X ∼ = T ∗ P n , soc( T X | P n ) = (1 + h ) n +1 (1 − h ) n +1 = (1 − h ) n +1 (cid:3) Let θ be the Beauville-Bogomolov class. Then for n = 4, Lemma 4.2. θ | P = − h .Proof. Let θ | P = nh . Equation (10) implies that 60 n = 7 · n ( − − − + 4(10) which implies thelemma. (cid:3) Finally, the last intersection theoretic piece of data we need is[ P ] = c ( N P /X ) = c ( T ∗ P ) = 5 (13)since P is Lagrangian.Assume now that X is deformation equivalent to a Hilbert scheme of 4 points on a K ℓ ∈ H ( X, Z ) be the class of the line, and λ = 6 ℓ ∈ H ( X, Z ), via the embedding H ( X, Z ) ⊂ H ( X, Z )induced by the Beauville-Bogomolov form. Note that λ | P = ( λ,λ )6 h since h λ | P , ℓ i = h λ, ℓ i = ( λ, λ ) by thedefinition of λ . Then[ P ] = aλ + bλ θ + cλ c ( X ) + dθ + eθ c ( X ) + f c ( X ) + gθα + h c ( X ) α Assume that α | P = yh , for y ∈ Q . Intersecting this class with each of (12), λ , λ θ, λ c ( X ) , θ , θ c ( X ) , c ( X ) , αθ, α c ( X )yields by Lemmas 4.1 and 4.2 the equation M ( λ )[ P ] = (cid:16) ( λ,λ )6 (cid:17) − (cid:16) ( λ,λ )6 (cid:17) − (cid:16) ( λ,λ )6 (cid:17) − y − y (14) rom which it follows that [ P ] = (cid:16) 25 + λ,λ ) + λ,λ ) (cid:17) − (cid:16) λ, λ ) + 3276 + λ,λ ) (cid:17) (cid:16) 23 + λ,λ ) (cid:17) (cid:0) ( λ, λ ) + 252( λ, λ ) − (cid:1) − (5( λ, λ ) − − y − y (15)Finally, (13) yields:5 = 25788299776 x + 17598537472 x + 40310948608 x − y + 733792 x + 6567584where x = ( λ, λ ). This may be rewritten as y = 5 · · x + 5 · x + 13 · · · x + 3 x − · · · (16)Note that while we may have y ∈ Q , x must be integral. Also note that there is a solution compatible withConjecture 2, namely ( x, y ) = ( − , Proposition 4.3. The only solution of (16) with x ∈ Z and y ∈ Q is ( x, y ) = ( − , . It then follows that Theorem 4.4. Let X be of K [4] -type, P ⊂ X be a smoothly embedded Lagrangian 4-plane, ℓ ∈ H ( X, Z ) the class of a line in P , and ρ = 2 ℓ ∈ H ( X, Q ) . Then ρ is integral, and [ P ] = 1337920 (cid:0) ρ + 1760 ρ c ( X ) − θ + 4928 θ c ( X ) − ( X ) (cid:1) (17) Further, we must have ( ℓ, ℓ ) = − .Proof. (17) is obtained from (15) by substituting ( λ, λ ) = − 126 and y = 0, after setting ρ = λ . It remainsto show that ρ is integral. Following [HHT], after deforming to a Hilbert scheme of points on a K S , we can write ℓ = D + mδ ∨ using the decomposition dual to (1), for D ∈ H ( S, Z ). Since( ℓ, ℓ ) = D − m − D ∈ Z , 3 | m . For 2 ℓ to be an integral class in H ( X, Z ), by Poincar´e duality it is sufficient for theform (2 ℓ, · ) on H ( X, Z ) to be integral, which it obviously is, since ( δ ∨ , δ ∨ ) = − . (cid:3) Solving the Diophantine equation The Diophantine equation (16) to solve is y = 5 · · x + 5 · x + 13 · · · x + 3 x − · · · with x ∈ Z and y ∈ Q . Let C be the affine curve described by the equation. After the change of variables( x , y ) = ( x + 126 , · · y ), every point ( x, y ) ∈ C with x ∈ Z gives an integral point ( x , y ) on the curve C : y = (5 · x − (2 · · ) x + (2 · · · · x − (2 · · · ) x emma 5.1. For an integer v consider the elliptic curve E v given by the Weierstrass equation y = x − (2 · · · v ) x + (2 · · · · · · v ) x − (2 · · · · · v ) Then every integral point ( x , y ) = (0 , on the curve C corresponds to an integral point ( x , y ) on one ofthe curves E v where x = u v x = 5 · · v u y = uvw y = 5 · · v w for some integers u, v, w where v is a divisor of · · · .Proof. Certainly if x = 0 then y = 0 and it can be checked that if y = 0 then x = 0 is the only rationalsolution. So let us assume for the remaining that x , y = 0. Note that since x ∈ Z it follows that y ∈ Z and x | y . Since x , y = 0 we may write x = u v and y = uvw for u, v, w ∈ Z with v square-free.Rewriting the equation we get vw = 5 · · u v − · · · u v + 2 · · · · · u v − · · · and we conclude that v is a divisor of 2 · · · · · v and making the change of variables y = 5 · · v · w and x = 5 · · v · u weget the equation(5 · · v · w ) = (5 · · v · u ) − · · · v · (5 · · v · u ) + 2 · · · · · · v (5 · · v · u ) − · · · · · v which yields y = x − · · · v · x + 2 · · · · · · v x − · · · · · v and therefore a point ( x , y ) ∈ E v ( Z ). (cid:3) Thus to find the required points on C we need to find the integral solutions of the elliptic curve E v abovewhenever v is a divisor of 2 · · · 11, of which there are 32 (positive and negative). Lemma 5.2. Suppose v is a divisor of · · · such that ∤ v . If the curve E v has an integral solution (5 · · u v , · · v w ) then v ∈ {− , − , − , − } .Proof. Note from the equation vw = 5 · · u v − · · · u v + 2 · · · · · u v − · · · we deduce that 7 | vw . Since 7 ∤ v it follows that 7 | w so it must be that 5 u v + 2 · · · u v ≡ u v ≡ u v (mod 7). Since v is invertible we get 5 u ≡ u v . If 7 ∤ u then we wouldhave that 5 is a quadratic residue mod 7, which is not true. So 7 | u . Rewriting the equation for w = 7 w and u = 7 u we get vw = 5 · · u v − · · · u v + 2 · · · · · u v − · · so necessarily vw ≡ v of 2 · · 11 for which such w exist are3 , , , , − , − , − , − | v then we could write v = 3 v so we would get v w = 5 · · · u v − · · · · u v + 2 · · · · · u v − · · which would imply that 3 | v w . Since 3 ∤ v (as v is square-free) it follows that 3 | v w but then 3 divides the right hand side so we deduce that 3 | u . Writing w = 3 w and u = 3 u we get v w = 5 · · · u v − · · · · u v + 2 · · · · · u v − · · As before, we get that 3 | v w but now 3 cannot divide the right hand side.The remaining possibilities for v are − , − , − , − (cid:3) Lemma 5.3. If the curve E v where v is a divisor of · · · such that | v has an integral solution (5 · · u v , · · v w ) then v ∈ { , , , } . roof. Writing v = 7 v we get v w = 5 · · u v − · · · u v + 2 · · · · · u v − · · · Since v is square-free 7 ∤ v so we deduce that 7 | w . Writing w = 7 w we get7 v w = 5 · · u v − · · · u v + 2 · · · · u v − · · which implies that u v ≡ v among the square-free divisors of 2 · · 11 for which such u exist are 1 , , , , − , − , − , − 66 giving v ∈ { , , , , − , − , − , − } .As in the previous lemma, under the assumption that 3 | v we get a contradiction. The remainingpossibilities are v ∈ { , , , } . (cid:3) Six of the eight cases to which we’ve reduced in Lemmas 5.2 and 5.3 are then treated directly by: Lemma 5.4. If v ∈ {− , − , , , , } the curve E v has no integral points of the form (5 · · u v , · · v w ) .Proof. We will compute the integral points of these elliptic curves using Sage ([S + geom.math.washington.edu . The general method is by finding a basis for theMordell-Weil group of a rational elliptic curve (using the command gens in Sage) and then finding a listof all the integral points using this basis (using the command integral points(mw basis=...) in Sage).Typically the computation of a basis is very difficult computationally (on the order of hours for the curvesunder consideration), whereas the computation of integral points is quite fast (on the order of seconds). Assuch we include bases for the Mordell-Weil groups of these elliptic curves in which case the computation ofintegral points can be reproduced quickly.Using Sage we find that the curves E (rank 1 with generator ( , ) found in 22 seconds), E (rank 1 with generator ( , ) found in 6 hours 45 minutes)and E (rank 1 with generator ( , − ) found in 6 hours 40 minutes) have no in-tegral points. Further computations show that the curves E (rank 4 with generators (564480 , , , , E − (rank 3 withgenerators ( − , / , / 8) and (166980 : 85186200) found in 1 hour 20 minutes)and E − (rank 2 with generators ( − , ) and ( − , − ) found in 1 hour18 minutes) have integral points, but none of them has the x -coordinate of the required form. Indeed, E − has 6 integral points, ( − , ± − , ± , ± x -coordinates are of the required form x = 5 · · ( − · u ; E − has two integral points (0 , ± x -coordinate was assumed to be nonzero; finally, E has 34 integral points (564480 , ± , ± , ± , ± , ± , ± , ± , ± , ± , ± , ± , ± , ± , ± , ± , ± , ± x -coordinates are of theform 2 · · · u . (cid:3) The remaining two curves E − , E − are computationally less tractable. The standard computation ofgenerators for the Mordell-Weil group in Sage for these two elliptic curves does not terminate in any reason-able time, though the closed-source algebra system Magma ([BCP97]) allows one to perform a reasonablyfast analysis of these two elliptic curves. We will give two computational proofs that these curves do nothave integral points of the required type: the first, in the open source Sage, relies on Kolyvagin’s proof of theBirch and Swinnerton-Dyer conjecture of elliptic curves over Q of analytic rank 1 while the second, in theproprietary Magma, uses a two descent procedure, and is given mainly as a corroboration of the results fromSage. We are greatful to Michael Stoll for explaining how to do the computations in Magma. We remarkthat the same methods will in principle work for the other curves in Lemma 5.4 of rank 1, namely E and E .We first need the following lemma. Lemma 5.5. If E is one of the curves E − and E − then L ′ ( E, = 0 . roof. We recall a result of Cohen ([Coh93, 5.6.12]) that L ′ ( E, 1) = 2 X n ≥ a n n E (cid:18) πn √ N (cid:19) where N is the conductor of E and E ( x ) = R ∞ e − xy y − dy is the exponential integral. Truncating thisseries at k , one gets L ′ ( E, 1) = L k + ε k where L k = 2 k X n =1 a n n E (cid:18) πn √ N (cid:19) and the error is explicitly bounded | ε k | ≤ e − π ( k +1) / √ N / (1 − e − π/ √ N ) (for a proof see [GJP + § E.lseries().deriv at1(k) (here k is the cutoff). In principle, if one expects that L ′ ( E, = 0then it suffices to choose the cutoff index k large enough that | ε k | < | L k | in which case L ′ ( E, 1) will be forcedto be nonzero.However, the curves under consideration have such a large conductor (in both cases N = 83060209520534400)that k has to be choosen on the order of 8 · , which is too large for practical purposes in Sage: in effect oneruns out of memory in the computation of the coefficients a n and E (2 πn/ √ N ). We compute the coefficients a n for the two curves up to k = 8 · by first computing a p for p prime (this operation takes about 2 hoursfor each curve) and then reconstructing a n using the following: if ( m, n ) = 1 then a mn = a m a n , if p ∤ N then a p k = a p a p k − − pa p k − and if p | N then a p k = a kp . For each curve the resulting file is on the order of 2 . GB and the computation takes about 3 hours for each curve. Next, we compute E (2 πn/ √ N ) for 1 ≤ n ≤ · (once, as the two curves have the same conductor). The command exponential integral 1( π/ √ N , k) in Sage should return the desired list but k is too large for this operation to be feasible. Instead, notingthat Sage’s exponential integral 1 is a wrapper for the PARI ([The12], version 2.5.4) function veceint1 ,we rewrote this PARI function to write the coefficients E (2 πn/ √ N ) to a file, instead of collecting them ina prohibitively long vector. The subsequent computation was run for about 10 hours resulting in 35GB ofdata.Each coefficient E (2 πn/ √ N ) = E ,n + ε ,n where E ,n is the number computed in PARI and | ε ,n | < − is the chosen precision. We denote by ℓ E the value 2 P kn =1 a n n E ,n computed in Sage and PARI using thecutoff k = 8 · . Therefore we compute the value of L ′ ( E, 1) = ℓ E + ε where the error is then at most(using the inquality | a n | ≤ n from [GJP + 09, Lemma 2.9]) ε < k X n =1 | a n | n · − + ε k < · − · k + ε k < · − + ε k < ℓ E − = 12 . . . . and ℓ E − = 16 . . . . and the conclusion follows. (cid:3) Lemma 5.6. If v ∈ {− , − } the curve E v has no integral points of the form (5 · · u v , · · v w ) .Proof. First, suppose E/ Q is an elliptic curve of rank 1 and P ∈ E ( Q ) is a point of infinite order (a factwhich can be checked computationally by requiring that the canonical height of the point is nonzero). Wewould like a fast algorithm for finding a generator P of the Mordell-Weil group E ( Q ). Suppose P is agenerator of E ( Q ) in which case P = nP for some integer n as E has rank 1. If P is not a generator then | n | ≥ h for the logarithmic height and b h for the canonical logarithmic height on E ( Q ). There exists aconstant B , depending only on E , called the Cremona-Pricket-Siksek bound, such that for all Q ∈ E ( Q ), h ( Q ) ≤ b h ( Q ) + B . Given a Weierstrass equation for E , the constant B can be computed in Sage usingthe command CPS height bound and in Magma using the command SiksekBound . If | n | ≥ b h ( P ) ≤ b h ( P ) n ≤ b h ( P )4 so h ( P ) = b h ( P ) + h ( P ) − b h ( P ) ≤ b h ( P )4 + B . Thus, to find P one only needs to search for ational points of height at most 14 b h ( P ) + B . One can find rational points of height ≤ h in Sage using thecommand rational points(bound= h ) and a generator P can be found in the resulting finite list.We will first check that the elliptic curves E − and E − have rank 1 and then we will apply the abovedescribed procedure to find a basis for the Mordell-Weil group. The command DescentInformation inMagma rapidly returns rank 1 for our curves. As mentioned above, in Sage one needs a different approach(note that the Sage command analytic rank yields only the probable analytic rank, equal to 1, in about 17hours for each of the two curves).Recall Kolyvagin’s result that if E is a (necessarily modular) rational elliptic curve of analytic rank 0 or 1then the Birch and Swinnerton-Dyer conjecture is true, i.e., the rank of the elliptic curve equals its analyticrank. We will exhibit below points of infinite order on each of the two elliptic curves and so their rank (andso also their analytic rank) is at least 1. Lemma 5.5 implies that L ′ ( E, = 0 and so their analytic rank,and therefore also their rank, must be 1, as desired.We proceed with finding bases for the Mordell-Weil groups. We start with the curve E = E − . Theelliptic curve E is y = x + 2 · · · · x + 2 · · · · · · · x + 2 · · · · Via the change of variables x = 4 x − , y = 8 y we get the minimal Weierstrass equation E ′ y = x − x + 1933249267 x + 116312127942837One may easily check that the point P = (cid:18) , (cid:19) is in E ′ ( Q ) (this point was found using Magma, but checking that it is a point on the curve is immediatewithout necessarily using a computer). The command height in Sage computes the canonical height to be b h ( P ) = 11 . . . . (and so P has infinite order) while the CPS bound is B = 11 . . . . .As explained before, we seek a generator of E ′ ( Q ). If P is not a generator then a generator will haveheight at most b h ( P ) / B . However, a computation in Sage shows that the only rational points with thisheight bound are 0 , ± P and so P must be a generator of E ′ ( Q ).Transfering back to E ( Q ) one obtains the generator ( x, y ) = ( − , E ( Q ). Using thecommand integral points in Sage to compute the integral points, inputting manually the basis for E ( Q ),one obtains that E ( Q ) has the integral points ( − , ± x = − E = E − is y = x + 2 · · · · x + 2 · · · · · · · x + 2 · · · · via the change of variables x = 16 x − , y = 64 y gives the minimal model E ′ y = x + x + 483312317 x + 14539257649013Again one may easily check that the point P = ( − , − E ′ ( Q ). It has canonical height b h ( P ) = 5 . . . . and thus it has infinite order. The CPS bound is computed to be B = 10 . . . . . Asbefore this allows one to show that P is a generator of E ′ ( Q ). The point P corresponds to the point( − , E ( Q ). Finally, using this basis in the computation of integral pointsin Sage yields that the only integral points are ( − , ± x cannot be − (cid:3) Appendix: Equivariant Localization For the sake of completeness we describe the well-known computation of the integrals Z S [ n ] δ k c µ ( S [ n ] )for S = P , P × P and δ = det O [ n ] by toric localization.First consider S = A , which has an action by G = G m via ( x, y ) ( αx, βy ) where α, β are the charactersobtained by projecting to each factor. The only fixed point is the origin (0 , G also acts on ( A ) [ n ] ; fixed oints are length n subschemes Z fixed by G . Thus, they must be supported on a fixed point (i.e. the origin),and the ideal I Z ⊂ A = C [ x, y ] must be generated by monomials. I Z is determined by the monomials x a y b left out of the ideal, which form a Young tableau with n boxes. Given such a Young tableau in the upperright quadrant, let ( i, b i − 1) for 0 ≤ i ≤ n − b i is the height of the i th column.A partition µ of n uniquely determines a Young tableau by arranging µ i columns of height i in descendingorder.For a space X with an action by G with isolated fixed points, we can compute integrals over X byrestricting to the fixed point locus using Bott localization: Z X ϕ = X p ∈ X G Z i ∗ p ϕ c top ( T p X )where ϕ ∈ H ∗ G ( X ), i ∗ p : H ∗ G ( X ) → H ∗ G ( X G ) ∼ = H ∗ ( X G ) ⊗ H ∗ G ([pt]) is the pull-back to a fixed point p ∈ X G .The Chern class is the equivariant Chern class of the G representation T p X .For example, consider S = A again. For a partition µ representing a fixed point p µ of X = ( A ) [ n ] , theChern polynomial is [ES87, Lemma 3.2] C ( µ ; α, β ) := X i t i c n − i ( T p µ X ) = Y ≤ i ≤ j ≤ n b j − − Y s = b j ( t +( i − j − α +( b i − − s − β )( t +( j − i ) α +( s − b i − ) β ) (18) O [ n ] restricted to a point of A [ n ] corresponding to a subscheme Z is canonically O Z , so setting f = c ( O [ n ] ), Z ( µ ; α, β ) := i ∗ p µ f = n X i =0 b i − X j =0 ( iα + jβ ) (19)6.1. The case S = P . Let G m act on [ x, y, z ] via [ αx, βy, z ]. There are three fixed points p = [0 , , , p =[0 , , p = [1 , , n subscheme Z of P will consist of a length n i subscheme Z i at p i with P n i = n . The tangent space at such a point is canonically T Z ( P ) [ n ] = M i T Z i ( P ) [ n i ] Note that at any point [ Z ] ∈ ( P ) [ n ] corresponding to a subscheme Z supported at p i , there is a G m -stableZariski neighborhood isomorphic to A [ n ] with torus action via ( αx, βy ), ( αβ − x, β − y ), ( βα − x, α − y ) for i = 0 , , µ of n will be three partitions ( µ , µ , µ ) such that | µ | + | µ | + | µ | = n ; 3-vector partitions of n classify fixed points p µ of X = ( P ) [ n ] . By the above, the tangent space at p µ has Chern polynomial X t i C n − i ( µ ; α, β ) = C ( µ ; α, β ) C ( µ ; α − β, − β ) C ( µ ; β − α, − α ) (20)Define C i ( µ ; α, β ) = c i ( T p µ X ). Also, Z ( µ ; α, β ) := i ∗ p µ f = Z ( µ ; α, β ) + Z ( µ ; α − β, − β ) + Z ( µ ; β − α, − α ) (21)The final answer is then, for X = ( P ) [ n ] Z X f n − P i k i Y i c k i ( T X ) = X µ, | µ | = n Z ( µ ; α, β ) n − P i k i Q i C k i ( µ ; α, β ) C n ( µ ; α, β ) (22)6.2. The case S = P × P . Let G m act on S = P × P via [ αx , y ] × [ βx , y ]. The fixed points areclassified by 4-vector partitions µ . Now we have X t i C ′ n − i ( µ ; α, β ) = C ( µ ; α, β ) C ( µ ; − α, β ) C ( µ ; α, − β ) C ( µ ; − α, − β ) (23)Also, Z ′ ( µ ; α, β ) := i ∗ p µ f = Z ( µ ; α, β ) + Z ( µ ; − α, β ) + Z ( µ ; α, − β ) + Z ( µ ; − α, − β ) (24) he final answer is then once again Z ( P × P ) [ n ] f n − P i k i Y i c k i ( T X ) = X µ, | µ | = n Z ′ ( µ ; α, β ) n − P i k i Q i C ′ k i ( µ ; α, β ) C ′ n ( µ ; α, β ) (25)6.3. Universal Polynomials. Let Φ be the universal genus from Section 3.1. We have X n ≥ z n Z S [ n ] exp det( O [ n ] )Φ( S [ n ] ) = A ( z ) c ( S ) B ( z ) c ( S ) We have computed explicitly in SAGE the power series A and B for vanishing odd Chern classes up todegree 20, and the result can be found on the authors’ webpages. 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Sage Mathematics Software (Version 5.2) . The Sage Development Team, 2013. .[The12] The PARI Group, Bordeaux. PARI/GP, version , 2012. available from http://pari.math.u-bordeaux.fr/ . Voi92] C. Voisin. Sur la stabilit´e des sous-vari´et´es Lagrangiennes des vari´et´es symplectiques holomorphes. Complex projectivegeometry , 179:294, 1992. B. Bakker: Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York,NY 10012 E-mail address : [email protected] A. Jorza: University of Notre Dame, 275 Hurley, Notre Dame, IN 46556 E-mail address : [email protected]@nd.edu