Blown-up toric surfaces with non-polyhedral effective cone
Ana-Maria Castravet, Antonio Laface, Jenia Tevelev, Luca Ugaglia
BBLOWN-UP TORIC SURFACES WITHNON-POLYHEDRAL EFFECTIVE CONE
ANA-MARIA CASTRAVET, ANTONIO LAFACE, JENIA TEVELEV, AND LUCA UGAGLIA
Abstract.
We construct examples of projective toric surfaces whose blow-upat a general point has a non-polyhedral pseudoeffective cone, both in charac-teristic 0 and in prime characteristic. As a consequence, we prove that thepseudo-effective cone of the Grothendieck–Knudsen moduli space M ,n of sta-ble rational curves is not polyhedral for n ≥
10 in characteristic 0 and incharacteristic p for an infinite set of primes of positive density. In particular, M ,n is not a Mori Dream Space in characteristic p in this range, which isalso a new result. Our analysis in characteristic p is based on applying toolsof arithmetic geometry and Galois representations to a very interesting classof arithmetic threefolds that we call arithmetic elliptic pairs of infinite order. Introduction
An effective cone of a projective variety X and its closure, the pseudo-effectivecone Eff( X ), contain an impressive amount of information about the birationalgeometry of X . An even finer invariant is the Cox ring Cox( X ), at least whenthe class group Cl( X ) is finitely generated. If X is a Mori Dream Space (MDS)then Cox( X ) is finitely generated, which in turn implies that Eff( X ) is polyhedral.A basic example of a MDS is a projective toric variety [Cox95]. Its effective cone isgenerated by classes of toric boundary divisors. For a toric variety P , we denote byBl e P its blow-up at the identity element of the torus. Our main result contributesto the growing body of evidence that this is a very intriguing class of varieties. Theorem 1.1.
There exist projective toric varieties P , including toric surfacesgiven by lattice polygons from Database 9.1, such that Eff(Bl e P ) is not polyhedral incharacteristic . For some of these toric surfaces (see Theorem 5.6), Eff(Bl e P ) isnot polyhedral in characteristic p for an infinite set of primes p of positive density.Remark . We checked that for every prime p < P such that Eff(Bl e P ) is not polyhedral in characteristic p (see Database 9.2). Perhapsone can find P for every p , but this seems out of reach with our methods.Our main application is to the birational geometry of the Grothendieck–Knudsenmoduli space M ,n of stable rational curves with n marked points. The study ofeffective cones of moduli spaces has a long history, starting with the pioneeringwork of Harris and Mumford [HM82], who used computations of effective divisorsto show that M g is not unirational for g (cid:29)
0, in fact is a variety of general type.Every boundary divisor is an extremal ray of Eff( M ,n ), in fact these divisors areexceptional, i.e., can be contracted by birational contractions. For example, M , Mathematics Subject Classification.
Key words and phrases.
Toric varieties, elliptic curves, moduli of curves. a r X i v : . [ m a t h . AG ] S e p s a degree 4 del Pezzo surface, and its boundary divisors form the Petersen graphof ten ( − M , ). Extremal rays of a different type for M , were found by Keel and Vermeire [Ver02]. Hassett and Tschinkel proved in[HT02] that Eff( M , ) is generated by the boundary and Keel–Vermeire divisors.A large class of exceptional divisors on M ,n was discovered by Castravet andTevelev [CT13]. They are parametrized by irreducible hypertrees, which can be ob-tained, for example, from bi-colored triangulations of the 2-sphere. Up to the actionof the symmetric group S n , this gives 1 , , , , , . . . new types of exceptionaldivisors on M ,n for n = 7 , , , , , . . . . Equations of these divisors appear as nu-merators of scattering amplitude forms for n particles in N = 4 Yang–Mills theoryin the work of Arkani-Hamed, Bourjaily, Cachazo, Postnikov and Trnka [ABC + n marked points on a smooth or stable algebraic curveunder a random meromorphic function uniformly distributed with respect to thetranslation-invariant volume form of the Jacobian.New extremal rays of Eff( M ,n ) were found by Opie [Opi16] disproving an over-optimistic conjecture from [CT13]. Further extremal rays were found by Doran,Giansiracusa, and Jensen [DGJ17]. Our second result explains this complexity. Theorem 1.3.
The cone
Eff( M ,n ) is not polyhedral for n ≥ , both in charac-teristic and in characteristic p for an infinite set of primes p of positive density,including all the primes up to . The moduli space M ,n is related to blown-up toric varieties via the notion ofa rational contraction , a dominant rational map X (cid:57)(cid:57)(cid:75) Y of projective varietiesthat can be decomposed into a sequence of small Q -factorial modifications [HK00]and surjective morphisms. Given a rational contraction, if Eff( X ) is a (rational)polyhedral cone then Eff( Y ) is also (rational) polyhedral (Lemma 2.2). By [CT15],there exist rational contractionsBl e LM n +1 (cid:57)(cid:57)(cid:75) M ,n (cid:57)(cid:57)(cid:75) Bl e LM n , where LM n is the Losev–Manin moduli space of chains of rational curves with 2heavy and n − M ,n has essentially the same birational geometry as theblow-up of a toric variety in the identity element of the torus. Moreover, a featureof LM n , noticed in [CT15] and proved in Theorem 8.1, is “universality” among allprojective toric varieties P . Specifically, for any P there exist rational contractions LM n (cid:57)(cid:57)(cid:75) P and Bl e LM n (cid:57)(cid:57)(cid:75) Bl e P for n large enough. In particular, if the cone Eff(Bl e P ) is not polyhedral thenEff(Bl e LM n ) (and therefore Eff( M ,n )) are not polyhedral either.A similar strategy was used in [CT15] to show that M ,n is not a MDS in charac-teristic 0 for n ≥ n ≥
10. For n ≤ M ,n is a MDS [HK00, Cas09]. So the only open cases in characteristic 0 are n = 7 , , n ≥ M ,n at one (very general) point has a non-polyhedraleffective cone [HY]. oric surfaces used in [CT15] were the weighted projective planes P ( a, b, c ).Of course Bl e P ( a, b, c ) has Picard number 2 and its effective cone is polyhedral.Nevertheless, Goto, Nishida and Watanabe [GNW94] proved (motivated by Cowsik’squestion in commutative algebra) that Bl e P ( a, b, c ) is not a MDS in characteristic 0for certain values of a, b, c by exhibiting a nef but not semi-ample line bundle. Butin characteristic p , Bl e P ( a, b, c ) is a MDS by Artin’s criterion [Art62] and thereforethis technique cannot be used. So the following corollary of Theorem 1.3 is new: Corollary 1.4. M ,n is not a MDS in characteristic p for n ≥ and for aninfinite set of primes p of positive density, including all the primes up to . In characteristic 0, we prove Theorem 1.1 in § §
10, where we describe theMagma package and databases used for some routine computer-aided calculations.We show that the blow-up X = Bl e P is an elliptic pair of infinite order, the notionintroduced in § elliptic pair ( C, X ) is aprojective rational surface X with log terminal singularities and a curve C containedin the smooth locus of X and such that p a ( C ) = 1 and C = 0. Much of thegeometry is encoded in the restriction map res : C ⊥ → Pic ( C ), where C ⊥ ⊆ Cl( X )is the orthogonal complement. The order of an elliptic pair is the order of res( C ).If the order is infinite and ρ ( X ) ≥ X ) is not polyhedral (Lemma 3.3). Toric elliptic pairs and the related notion of good polygons are introduced in § P in 9 general points. By contrast, toric elliptic pairs are defined overthe base field. In particular, their order is automatically finite in characteristic p .We use log-MMP to construct a ( K + C )-minimal model ( C, Y ) of any ellipticpair (
C, X ). We focus on the study of polyhedrality of Eff( Y ). Of course if Eff( Y )is not polyhedral then Eff( X ) is also not polyhedral. Remarkably, Y has Du Valsingularities if the order is infinite (Corollary 3.12). On the other hand, if the orderis finite and Y has Du Val singularities, there is a simple criterion for polyhedrality(Corollary 3.17) in terms of the restriction map and the root sublattice T ⊂ E .The synthesis is given by the notion of an arithmetic elliptic pair of infinite order ,a flat pair of schemes ( C , X ) over an open subset in the spectrum of a ring ofalgebraic integers with elliptic pairs as geometric fibers, of infinite (resp. finite)order over an infinite (resp. finite) place. In § C , X ) of infinite order. Here we call aprime p polyhedral if Eff( Y ) is polyhedral, where ( C, Y ) is the minimal model ofthe geometric fiber (
C, X ) in characteristic p . This distribution is an intriguingquestion of arithmetic geometry, which we reduce to the question about reductionsof points on the elliptic curve in the spirit of the Lang and Trotter analysis [LT77].We prove Theorem 1.1 and Remark 1.2 in § § § §
10 involves the step ofcomputing a member of a linear system on the toric surface P that has a point oflarge multiplicity at the identity of the torus. We have very little control of thiscurve and prove its properties (irreducibility, etc.) by a computer-aided calculation.It’s not clear how to apply this implicit method in an infinite sequence of examples.By contrast, in § parametric method . We start with an elliptic curve C with points and describe its map to P that folds many points of C onto a pointof high multiplicity. We do this for an infinite sequence of toric surfaces P k and asequence of elliptic curves C k that are themselves members of an elliptic fibration. inally, in § C, X ) with X smooth, of largePicard number ρ = 18 and such that the Mordell–Weil rank of C is equal to 9. Wedon’t know if there is an upper bound on the Picard number of a toric elliptic pair. Acknowledgements.
A.L. has been partially supported by Proyecto FONDE-CYT Regular n. 1190777. J.T. was partially supported by the NSF grant DMS-1701704, Simons Fellowship, and by the HSE University Basic Research Programand Russian Academic Excellence Project ’5-100’.We would like to thank Igor Dolgachev and Brian Lehmann for useful discus-sions and answering our questions. We are especially grateful to Tom Weston forsharing and explaining his paper [Wes03]. In the REU directed by J.T. in 2017,Stephen Obinna [Obi17] has started to collect evidence for existence of blown-uptoric surfaces with non-polyhedral effective cone. The software Magma [BCP97]was used extensively. Some of the graphics are by the Plain Form Studio.
Contents
1. Introduction 12. Polyhedrality of effective cones 43. Elliptic pairs: general theory 74. Good polygons and toric elliptic pairs 125. Arithmetic toric elliptic pairs of infinite order 156. Infinite sequences of good polygons 237. A smooth toric elliptic pair 328. On the effective cone of M ,n Polyhedrality of effective cones
Let k be an algebraically closed field of arbitrary characteristic. We recall somedefinitions (see for example [Laz04a, Laz04b]). If X is a normal projective irre-ducible variety over k , let Cl( X ) be the divisor class group and let Pic( X ) be thePicard group of X . As usual, we denote by ∼ the linear equivalence of divisors andby ≡ the numerical equivalence. Recall that for Cartier divisors D , D , we have D ≡ D if and only if D · C = D · C , for any curve C ⊆ X . We letNum ( X ) := Pic( X ) / ≡ be the group of numerical equivalence classes of Cartier divisors on X . We denoteNum ( X ) R = Num ( X ) ⊗ Z R , Num ( X ) Q = Num ( X ) ⊗ Z Q .Sometimes we extend ∼ to the linear equivalence of Q -divisors in a usual way(for Q -divisors, A ∼ B if kA ∼ kB as Cartier divisors for some k >
0) but mostlywe use numerical equivalence of Q -divisors to avoid confusion.Similarly, we define Z ( X ) R to be the group of R -linear combinations of irre-ducible curves in X , i.e., formal sums γ = (cid:88) a i C i , a i ∈ R ith all C i ⊆ X irreducible curves. As in [Laz04a, Def. 1.4.25], we letNum ( X ) R = Z ( X ) R / ≡ , where for two one-cycle classes γ , γ ∈ Z ( X ) R we have numerical equivalence γ ≡ γ if and only if D · γ = D · γ for all Cartier divisors D on X . It followsfrom the definitions thatNum ( X ) R ⊗ Num ( X ) → R , ( δ, γ ) (cid:55)→ δ · γ ∈ R is a perfect pairing, so Num ( X ) R and Num ( X ) R are dual vector spaces. In par-ticular, both Num ( X ) R and Num ( X ) R are finite dimensional real vector spaces.We define the pseudo-effective coneEff( X ) ⊆ Num ( X ) R , as the closure of the effective cone Eff( X ), i.e., the convex cone generated by nu-merical classes of effective Cartier divisors ([Laz04b, Def. 2.2.25]). We let Nef( X ) ⊆ Num ( X ) R be the cone generated by the classes of nef divisors . We defineMov ( X ) ⊆ Num ( X ) R the closure of the cone generated by numerical classes of movable X ) and Mov ( X ) are dual to each other. Thiswas proved first in [BDPP13] for the case when X is a smooth projective variety incharacteristic 0, but it holds in general. For X irreducible projective variety over afield k of characteristic 0 this is proved in [Laz04a, Thm. 11.4.19]. For the case ofarbitrary characteristic, the same proof holds, see for example [Ful17, Rmk. 2.1]. Definition 2.1.
A convex cone
C ⊆ R s is called polyhedral if there are finitelymany vectors v , . . . v s ∈ R s such that C = R ≥ v + . . . + R ≥ v s . The cone is saidto be rational polyhedral if one can choose the v i ’s in Q s . Lemma 2.2.
Let f : X → Y be a surjective morphism of normal projective irre-ducible varieties. If Eff( X ) is (rational) polyhedral then the same is true for Eff( Y ) .Proof. Suppose Eff( X ) is a (rational) polyhedral cone. By the duality between thecones Eff( X ) and Mov ( X ), it follows that Mov ( X ) is also a (rational) polyhedralcone. The proper push-forward of 1-cycles induces a map of R -vector spaces f ∗ : Num ( X ) R → Num ( Y ) R . By [FL17, Cor. 3.12], f ∗ (Mov ( X )) = Mov ( Y ). The definitions of Num ( X )and Mov ( X ) given in [FL17] coincide with the ones given above, see [FL17, Sec-tion 2.1, Ex. 3.3]. It follows that Mov ( Y ) is a (rational) polyhedral cone. Againby the duality between the cones Eff( Y ) and Mov ( Y ), it follows that Eff( Y ) is a(rational) polyhedral cone. (cid:3) We concentrate on the case of surfaces. The cone and contraction theorems holdin any characteristic with very mild assumptions, see [KM98, Tan14, FT12, Fuj20].
Proposition 2.3.
Let X be a normal projective Q -factorial surface with Picardnumber at least and such that the cone Eff( X ) is polyhedral. Then: (1) Every class C ∈ Num ( X ) of self intersection (or its opposite − C ) is inthe relative interior of either the cone Eff( X ) or its codimension one facet. The effective cone
Eff( X ) is generated by finitely many negative curves .In particular, Eff( X ) = Eff( X ) is a rational polyhedral cone. Part (2) of Proposition 2.3 appears also in [Nik00].
Proof. (1) Fix h an ample divisor. Let Q := { ω | ω ≥ , ω · h ≥ } ⊆ Num ( X ) R be the non-negative part of the light cone. Then either C or its opposite − C lieson the boundary ∂Q . By Riemann-Roch, the cone Q is contained in Eff( X ). Sincethe Picard number of X is at least 3, the cone Q is round. In particular, ∂Q canintersect only a facet of Eff( X ) of codimension 1 and only in its relative interior.(2) By (1), any ω ∈ Num ( X ) generating an extremal ray of Eff( X ) has ω < , for any such ω there exists an irreducible curve E suchthat ω is a positive multiple of the class of E . (cid:3) Proposition 2.4.
Let X be a normal projective Q -factorial surface with Picardnumber at least . Assume that C ⊆ X is an irreducible curve with C = 0 and C ≡ − αK X with α ∈ Q > . Then the following are equivalent: (1) There exist irreducible curves B , . . . , B s , with classes not equal to a rationalmultiple of the class of C , generating C ⊥ ⊆ Num ( X ) Q , and such that C ≡ a B + . . . + a s B s with a , . . . a s ∈ Q > . (2.1)(2) Eff( X ) is a rational polyhedral cone generated by negative curves.Proof. Proposition 2.3 gives (2) ⇒ (1). We prove the converse under our additionalassumptions. Note that C (hence, − K ) is nef. Recall that any ω ∈ Num ( X ) R generating an extremal ray must have ω ≤ ω < ω is the class ofa multiple of a curve [Deb01, Lemma 6.2]. In our set-up, the same is true when ω = 0. Indeed, if ω · C = 0, by the Hodge Index theorem, ω and C generate thesame ray. If ω · C >
0, then ω · K < ω is generated by the class of a curveby the Cone theorem. Hence, it suffices to prove that X contains finitely manyirreducible curves E with E ≤ E is not numerically equivalent to arational multiple of C . We can also assume that E (cid:54) = B i for all i .We consider two cases. If E · C = 0 then E · B i = 0 for all i by (2.1) and by ourassumption that E (cid:54) = B i for all i . Since B . . . , B s generate C ⊥ over Q , E must benumerically equivalent to a rational multiple of C , which we have also ruled out.Suppose E · C >
0. Since B . . . , B s generate C ⊥ over Q , the classes whichhave fixed intersections with the B i ’s form an affine subspace of dimension onein Num ( X ) Q , differing one from another by a multiple of the class of C . Since E · C > C · C = 0, there is at most one such class with E also fixed. Hence,it suffices to prove that E · B i and E belong to a finite set. By assumption (1) andadjunction, we have 1 α (cid:88) a i ( E · B i ) = E · ( − K ) ≤ E + 2 ≤ . Hence, 0 ≤ E · B i ≤ α/a i . As there exists l ∈ Z > (the index of Pic( X ) in Cl( X ))such that the lD is Cartier for any curve D (hence, l ( D · E ) is an integer), it follows A negative curve is an irreducible curve with negative self-intersection. The proof in [Deb01] is for smooth surfaces but the argument works verbatim in our case. hat E · B i belongs to a finite set. We have − ≤ E by adjunction and nefnessof − K . As E ≤
0, it follows similarly that E must belong to a finite set. (cid:3) Elliptic pairs: general theory
As in §
2, we work over an algebraically closed field k of arbitrary characteristic.While Propositions 2.3 and 2.4 address polyhedrality of Eff( X ) for a general sur-face X , in this section we study polyhedrality further for a rational surface in thepresence of a curve C with self-intersection 0 under some additional assumptions. Definition 3.1. An elliptic pair ( C, X ) consists of a projective rational surface X with log terminal singularities and an arithmetic genus one curve C ⊆ X disjointfrom the singular locus of X and such that C = 0. Let C ⊥ ⊆ Cl( X ) be theorthogonal complement to C . We define the restriction map res : C ⊥ → Pic ( C ) , D (cid:55)→ O ( D ) | C . Since K · C = 0 by adjunction, we can also define the reduced restriction map res : Cl ( X ) := C ⊥ / (cid:104) K (cid:105) → Pic ( C ) / (cid:104) res( K ) (cid:105) . We will often study a birational morphism X → Y , which is an isomorphism in aneighborhood of C . We will then use notation C X , C Y , etc, to avoid confusion.The most familiar elliptic pairs are rational elliptic fibrations X → P with afiber C (which can be a multiple fiber). However, we do not make this assumption.Note that as X is rational, h ( X, O X ) = 0, and hence Pic( X ) Q = Num ( X ) Q . Lemma–Definition 3.2.
We define the order e = e ( C, X ) of the elliptic pair ( C, X ) to be the positive integer satisfying any of the following equivalent conditions(or ∞ if none of them are met): (1) res( C ) ∈ Pic ( C ) is a torsion line bundle of order e . (2) e is the smallest positive integer such that h ( C, res( eC )) = 1 . (3) e is the smallest positive integer such that h ( X, eC ) = 2 . (4) e is the smallest positive integer such that h ( X, eC ) > .The order e ( C, X ) only depends on a Zariski neighborhood of C in X .Proof. The equivalence of (1) and (2) is clear. We use this as a definition of e .In particular, e ( C, X ) only depends on a Zariski neighborhood of C in X . Sincelog terminal singularities are rational and C is disjoint from the singular locusof X , if ˜ X is a resolution of singularities of X , then h ( ˜ X, nC ˜ X ) = h ( X, nC X )for any integer n . Hence, to prove the remaining equivalences we may assume that X is smooth. For any n ≥
0, we have h ( X, nC ) = h ( X, K X − nC ) = 0, sinceotherwise K X would be effective. Moreover, by Riemann-Roch χ ( O X ( nC )) = 1for all n . Thus either h ( X, nC ) = 1 and h ( C, res( nC )) = 0 for every n >
0, orfor some n > h ( X, nC ) = 2, h ( C, res( nC )) = 1 and h ( X, lC ) = 1, h ( C, res( lC )) = 0 for 1 ≤ l < n . (cid:3) Lemma 3.3.
Suppose ( C, X ) is an elliptic pair. Let e = e ( C, X ) . Then (1) e < ∞ if and only if C is a (multiple) fiber of a (quasi)-elliptic fibration . (2) If e = ∞ , then C is rigid, which means that h ( nC ) = 1 for all n > .In this case Eff( X ) is not polyhedral if the Picard number ρ ( X ) ≥ . If C is smooth or if char k (cid:54) = 2 , roof. Suppose e < ∞ . Then eC ∼ (cid:80) D i , for some irreducible curves D i (cid:54) = C byLemma 3.2 (3). As C = 0, it follows that the D i ’s are disjoint from C and | eC | is a base-point-free pencil. Since C = K · C = 0 by adjunction, ϕ | eC | : X → P is a (quasi)-elliptic fibration. Suppose e = ∞ . Then C is rigid by Lemma 3.2 (4).By Prop. 2.3, if Eff( X ) is polyhedral and the Picard number of X is at least 3, thenEff( X ) is generated by negative curves and C is contained in the interior of a facet.Thus h ( X, kC ) > k and therefore e ( C, X ) < ∞ by Lemma 3.2 (4). (cid:3) Lemma 3.4. If ( C, X ) is an elliptic pair, then K X + C is an effective divisor.Proof. As C is contained in the smooth locus of X , we can pass to a resolutionof singularities and prove for a smooth surface X that h ( − C ) = h ( K + C ) > O X ( K + C ) | C (cid:39) ω C (cid:39) O C , so there is an exact sequence → O X ( K ) → O X ( K + C ) → O C → . The statement follows from the vanishing h ( X, K ) = h ( X, K ) = 0. (cid:3)
Definition 3.5.
We say that (
C, X ) is a minimal elliptic pair if it does not containirreducible curves E such that K · E < C · E = 0. Remark . A curve E as in the definition must have E <
0. Indeed, E ≤ C and E are multiples of each other. But since E · K < C · K = 0, the latter is notpossible. Moreover, E is a rational curve [Fuj20, Thm 5.6]. By the contractiontheorem, there exists a morphism φ : X → Y contracting only E . As φ is anisomorphism in a Zariski neighborhood of C and Y is log terminal, ( C, Y ) is anelliptic pair. Moreover, K X ≡ φ ∗ K Y + aE , for some a ∈ Q . Since E · K X < E <
0, it follows that a >
0. Furthermore, K X < K Y . Lemma 3.7.
Let ( C, X ) be an elliptic pair. The following conditions are equivalent: (1) ( C, X ) is minimal; (2) K + C is nef; (3) C ∼ α ( − K ) , for some α ∈ Q > , a linear equivalence of Q -divisors; (4) K = 0 .Proof. To prove (1) ⇒ (2), assume that K + C is not nef. By the cone theorem for a log surface ( X, C ) [Tan14, Fuj20], there is an irreducible curve E such that( K + C ) · E < E <
0. Since K + C is effective, E must be one of itscomponents. Since C · ( K + C ) = 0 and C is nef, we must have C · E = 0, hence, K · E <
0. This contradicts the minimality of (
C, X ).Next we prove (2) ⇒ (3). Since ( K + C ) · C = 0, by the Hodge Index theoremwe must have ( K + C ) ≤
0. But since K + C is nef, ( K + C ) ≥
0. Thus( K + C ) = 0 and it must be that K + C ≡ λC , for some λ ∈ Q . As no multiple of K is effective, it follows that C ≡ α ( − K ), for some α ∈ Q > . Since X is rational,in fact C ∼ α ( − K ), a linear equivalence of Q -divisors;The implication (3) ⇒ (4) is clear. To see (4) ⇒ (1), suppose ( X, C ) is notminimal. By Remark 3.6, there is a contraction φ : X → Y of a curve E such that K · E < E < C · E = 0. Moreover, K Y > K X = 0. But ( C, Y ) is an ellipticpair and so K Y ≤ (cid:3) This trick is from the proof of the canonical bundle formula for elliptic fibrations in [BM77] Note that there are no singularity assumptions on K + C in the cone theorem for surfaces. heorem 3.8. Let ( C, Z ) be an elliptic pair with smooth Z . Then ( C, Z ) is mini-mal if and only if ρ ( Z ) = 10 , or equivalently, K = 0 . If ( C, Z ) is minimal then (i) C ∼ n ( − K ) for some positive integer n ; (ii) The lattice Cl ( Z ) (cid:39) E (negative definite).Suppose that ( C, Z ) is minimal and e ( C, Z ) < ∞ . The following are equivalent: (1) Eff( Z ) is polyhedral and generated by ( − and ( − curves. (2) Eff( Z ) is polyhedral. (3) Ker(res) contains linearly independent roots of E .Proof. By Lemma 3.7, the elliptic pair (
C, Z ) is minimal if and only if K = 0.Since Z is a smooth rational surface, it is an iterated blow-up of P or a Hirzebruchsurface F e . As K goes down by one and the Picard number goes up by onewhen blowing-up a smooth point, it follows that K = 0 if and only if ρ ( Z ) = 10.As Pic( Z ) has a Z -basis that includes the exceptional divisor D of the last blow-up, − K is a primitive vector of Pic( Z ) (its coefficient at D is equal to 1). It followsby Lemma 3.7 that C ∼ n ( − K ) for some integer n >
0. For the same reason, thegroup Cl ( Z ) has no torsion. By the Hodge Index theorem, the intersection formon Z makes Cl ( Z ) into an even negative-definite lattice of rank 8.We prove that the lattice Cl ( Z ) is unimodular, therefore isomorphic to E .We claim that the intersection pairing on Pic( Z ) is the same as when Z is theblow-up of P in 9 points (and hence, K ⊥ / (cid:104) K (cid:105) = E ). Assume that the minimalmodel of Z is F e . Let g , f be a basis of Pic( F e ) with g = − e , g · f = 1, f = 0.If e = 2 d + 1 is odd, then the intersection lattice on Pic( F e ) is the same as that ofa blow-up of P in one point, for example by considering the basis g + ( d + 1) f , g + df . If e = 2 d is even, then the intersection lattice on Pic( F e ) is the same asthat of P × P , for example given by the basis g + df , f . Thus the intersectionpairing on the blow-up of F e at one point is the same as on the blow-up of P intwo points. This finishes the proof of the claim.Suppose that ( C, Z ) is minimal and e = e ( C, Z ) < ∞ . By Lemma 3.3, | eC | givesa (quasi)-elliptic fibration Z → P . Clearly (1) ⇒ (2) and Prop. 2.3 (2) implies(2) ⇒ (1), as the only negative curves are ( −
1) and ( −
2) curves when Z is smoothand − K is nef. Assume (1). By Proposition 2.4, C ≡ (cid:80) a i B i for a i ∈ Q > , withirreducible curves B i different from C generating C ⊥ over Q . Since B i is irreducible,res( B i ) = 0. Since B i · K = 0, each B i is a ( −
2) curve. Since the curves B i generate C ⊥ over Q , eight of them are linearly independent modulo K . This proves (3).Assume (3). Let β , . . . , β be ( − C ⊥ , linearly independent mod-ulo K , and such that res( β i ) = 0. Adding to each β i a multiple of K , we mayassume that each β i restricts trivially to C . We claim that, for each i , either β i or( K + C ) − β i is effective. Indeed, for each β := β i we have a short exact sequence0 → O ( β − C ) → O ( β ) → O C → . If β is not effective, β − C is not effective either. Hence, h ( Z, O ( β − C )) >
0. But χ ( O ( β − C )) = 0 by Riemann–Roch. Thus h ( Z, O ( β − C )) > K + C ) − β iseffective. We have found 8 effective divisors D , . . . , D with res( D i ) = 0, D i = − K . Since each of the divisors D i belongs to a unionof the fibers of the (quasi)-elliptic fibration, there is an effective divisor D (cid:48) i (cid:54) = 0 suchthat D i + D (cid:48) i ∼ n i C for some n i . Hence, lC ∼ (cid:80) ( D i + D (cid:48) i ) for some l (cid:29)
0. Allcomponents of reducible fibers of the (quasi)-elliptic fibration | eC | (in particularall components of D i , D (cid:48) i ) are ( −
2) curves (in particular, not equal to C ). To onclude that Eff( Z ) is a polyhedral cone by Prop. 2.4 (1), we only need to checkthat components of D i , D (cid:48) i span C ⊥ over Q . But this is clear: a non-zero multipleof K is in their linear span and D , . . . , D are linearly independent modulo K . (cid:3) Remark . The proof of Theorem 3.8 shows that another condition equivalent to(1)–(3) is existence of effective divisors D , . . . , D , linearly independent modulo K (or equivalently C ), with D i · C = 0 for all i . The existence of such divisors ifEff( Z ) polyhedral is part of the proof of the theorem. Conversely, if such divisorsexist, we may assume w.l.o.g. that no components of D i equal C . Then argumentin the proof of Theorem 3.8 shows that condition (1) in Proposition 2.4 is satisfied. Theorem 3.10.
For any elliptic pair ( C, X ) , there exists a minimal elliptic pair ( C, Y ) and a morphism π : X → Y , which is an isomorphism over a neighbor-hood of C . Consider the Zariski decomposition on X of K + C , K + C ∼ N + P, N = a C + . . . + a s C s , a i ∈ Q > , the linear equivalence of Q -divisors . Then: (1) Y is obtained by contracting curves C , . . . , C s on X . (2) P ≡ if and only if − K Y ∼ C Y , in which case N is an integral combinationof C , . . . , C s and Y has Du Val singularities. Definition 3.11.
We call an elliptic pair (
C, Y ) a minimal model of (
C, X ). Corollary 3.12.
Let ( C, Y ) be a minimal model of an elliptic pair ( C, X ) such that e ( C, X ) = ∞ . Then Y has Du Val singularities. Consider the Zariski decomposition K + C ∼ N + P on X . Then K + C ∼ N is an integral effective combination of irreducible curves C , . . . , C s with a negative-definite intersection matrix. The minimal model Y isobtained by contracting curves C , . . . , C s and C Y ∼ − K Y .Proof. We first prove the theorem and then its corollary. We obtain a minimalmodel π : X → Y by running a ( K + C )-MMP [Tan14, Fuj20]. Equivalently(by Lemma 3.7), π is a composition of contractions of the form φ : X → Y , whereeach φ is the contraction of a K -negative curve E such that E · C = 0. On each step, K X + C X ∼ φ ∗ ( K Y + C Y ) + aE , with a ∈ Q > , a linear equivalence of Q -divisors.At the end we obtain that K Y + C Y is nef, i.e., ( C, Y ) is minimal. If the curvescontracted by π are C , . . . , C s ⊆ X , then K X + C X ∼ N + P , with P = π ∗ ( K Y + C Y ) , N = s (cid:88) i =1 a i C i a i ∈ Q > , a linear equivalence of Q -divisors. The divisor P is nef and effective (Lemma 3.4)and P · C i = 0 for all i . Hence, this is the Zariski decomposition of K + C . Moreover, P ≡ K Y + C Y ∼ P ≡
0. Recall an algorithm for computing the Zariski decom-position [Bau09]. Write K + C ∼ b B + . . . + b t B t as an integral, effective sum ofirreducible curves B i . Let N (cid:48) := (cid:80) x i B i , where 0 ≤ x i ≤ b i are maximal such that P (cid:48) := (cid:80) ( b i − x i ) B i intersects all C i non-negatively. Then N (cid:48) and P (cid:48) give a Zariskidecomposition of K + C . Since N = N (cid:48) is unique and P (cid:48) ≡ P ≡
0, the Zariski Recall that the C i ’s are irreducible curves with a negative-definite intersection matrix and P is a nef effective Q -divisor such that P · C i = 0 for all i . The Q -divisor N is determined uniquely. ecomposition is K + C ∼ b B + . . . + b t B t . To prove the singularity statement,note that − K Y ∼ C Y implies that K Y is Cartier. Thus Y has Du Val singularities.Finally, we prove the corollary. Suppose that e ( C, X ) = e ( C Y , Y ) = ∞ . If P (cid:54)≡ C Y ∼ α ( − K Y ), for some α ∈ Q , α (cid:54) = 1. Then C Y ∼ αα − ( K Y + C Y ), a linearequivalence of Q -divisors. But K Y + C Y restricts trivially to C Y by adjunction,and therefore res( C ) is torsion, which is a contradiction. So we must have P ≡ (cid:3) Remark . We give an example of a minimal rational elliptic fibration that doesnot satisfy C ∼ − K . Let W be a minimal smooth rational elliptic fibration with anodal fiber I . Blow-up the node of the fiber and contract the proper transform ofthe fiber (which has self-intersection − Y with a (1 ,
1) singularity, which is log terminal. The fiber C throughthe singularity is a nodal multiple fiber of multiplicity 2. We have C ∼ C ∼ − K . Lemma 3.14.
Let ( C, Y ) be an elliptic pair such that Y has Du Val singularities.Let π : Z → Y be its minimal resolution. (1) ( C, Y ) is minimal if and only if ( C, Z ) is minimal. Equivalently, ρ ( Y ) = 10 − R, where R is the rank of the root system associated with singularities of Y . (2) Assume ( C, Y ) is a minimal elliptic pair and ρ ( Y ) ≥ . Then the followingare equivalent: • Eff( Y ) is a polyhedral cone; • Eff( Y ) is a rational polyhedral cone; • Eff( Z ) is a polyhedral cone.Proof. As K Z = π ∗ K Y , the pair ( C, Z ) is minimal if and only if (
C, Y ) is minimalby Lemma 3.8. As ρ ( Y ) = ρ ( Z ) − R , the first statement follows. If Eff( Z ) is(rational) polyhedral then Eff( Y ) is (rational) polyhedral by Lemma 2.2. Assumenow Eff( Y ) is polyhedral. Then e ( C, Y ) < ∞ . By Proposition 2.3, C Y is containedin the interior of a maximal facet. It follows that C ⊥ Y contains ρ ( Y ) − K Y and restrict trivially to C . AsCl( Z ) Q decomposes as π ∗ Cl( Y ) Q ⊕ T Q , where T is a sublattice spanned by classesof ( − Y , we have ( C ⊥ Z ) Q = ( π ∗ C ⊥ Y ) Q ⊕ T Q . It followsthat C ⊥ Z contains ρ ( Y ) − R = 8 effective divisors which are linearly independentmodulo K Z and restrict trivially to C . It follows by Remark 3.9 that Eff( Z ) ispolyhedral. Thus Eff( Z ) is a rational polyhedral cone by Theorem 3.8. (cid:3) Definition 3.15.
Let (
C, X ) be an elliptic pair such that the minimal model (
C, Y )has Du Val singularities. Let π : Z → Y be the minimal resolution of Y . Let T ⊆ E = Cl ( Z )be a root sublattice spanned by classes of ( − Y .We call T the root lattice of ( C, X ). We denote by ˆ T its saturation E ∩ ( T ⊗ Q ).The push forward π ∗ : Cl( Z ) → Cl( Y ) induces a map Cl ( Z ) → Cl ( Y ) withkernel T , i.e., Cl ( Y ) (cid:39) E /T and the map res Z factors through res Y . Moreover,Cl ( Y ) / torsion (cid:39) E / ˆ T .
The intersection pairing on Y and pull back of Q -divisors realizes E / ˆ T as a sub-lattice of the vector space ( T ⊗ Q ) ⊥ ⊆ E ⊗ Q with the intersection pairing on Z . emark . Root lattices T ⊂ E were classified by Dynkin [Dyn57, Table 11].The quotient group Cl ( Y ) (cid:39) E /T was computed, e.g., in [OS91]. Corollary 3.17.
Let ( C, Y ) be a minimal elliptic pair with Du Val singularities and ρ ( Y ) ≥ . Let R be the rank of the root lattice of ( C, Y ) and suppose e ( C, Y ) < ∞ .Then Eff( Y ) is polyhedral if and only if there are roots β , . . . , β − R ∈ E \ ˆ T ,linearly independent modulo ˆ T and such that res( β i ) = 0 . In particular, if R = 7 then Eff( Y ) is polyhedral if and only if res( β ) = 0 for some root β ∈ E \ ˆ T .Remark . Let (
C, Y ) be a minimal elliptic pair with Du Val singularities and ρ ( Y ) = 2, the smallest possible. The root lattice of ( C, Y ) has rank R = 8.The effective cone Eff( Y ) is a rational polyhedral cone by the Cone theorem (it isspanned by the class of C and by the class of the unique negative curve). Thisdoesn’t provide any information about e ( C, Y ).The reader will notice a discrepancy between Corollary 3.17, which provides aneffective criterion of polyhedrality for minimal elliptic pairs (
C, Y ) with Du Valsingularities and e ( C, Y ) < ∞ and Corollary 3.12, which shows that a minimalmodel ( C, Y ) of an elliptic pair (
C, X ) with e ( C, X ) = ∞ has Du Val singularities.These disjoint scenarios are reconciled in the following definition: Definition 3.19.
Let (
C, X ) be an elliptic pair with e ( C, X ) = ∞ defined over K ,a finite extension of Q . Let R ⊂ K be the corresponding ring of algebraic integers.There exists an open subset U ⊂ Spec R and a pair of schemes ( C , X ) flat over U ,which we call an arithmetic elliptic pair of infinite order , such that • Each geometric fiber (
C, X ) of ( C , X ) is an elliptic pair of order e b whichdepends only on the corresponding point b ∈ U . We have e b < ∞ for b (cid:54) = 0. • The contraction morphism X → Y to the minimal model extends to thecontraction of schemes X → Y flat over U . • All geometric fibers (
C, Y ) of ( C , Y ) over U are minimal elliptic pairs withDu Val singularities and the same root lattice T ⊂ E .Let X , Y be geometric fibers over a place b ∈ U , b (cid:54) = 0. We call b a polyhedral prime if Eff( Y ) is polyhedral. If b is not polyhedral then Eff( X ) is also not polyhedral.Distribution of polyhedral primes is an intriguing question in arithmetic geom-etry that we will start to address for arithmetic toric elliptic pairs.4. Good polygons and toric elliptic pairs
At the beginning we work over an algebraically closed field k of any characteristic.We recall that a polygon ∆ ⊆ R is called a lattice polygon if its vertices are in Z .If ∆ is a lattice polygon, we will denote by Vol(∆) its normalized volume , i.e. twiceits euclidean area (so that Vol(∆) is always a non-negative integer). We recall thatgiven any Laurent polynomial f = (cid:88) u ∈ Z α u x u ∈ k [ x ± , x ± ] , (4.1)where x u := x u x u , we can construct a lattice polygon NP( f ), called the Newtonpolygon of f , by taking the convex hull of the points u ∈ Z such that α u (cid:54) = 0.A lattice polygon ∆ defines a morphism g ∆ : G m → P | ∆ ∩ Z |− , x (cid:55)→ [ x u : u ∈ ∆ ∩ Z ] , here x = ( x , x ) ∈ ( k ∗ ) . We will denote by P ∆ the projective toric surface defined by ∆, i.e. the closure of the image of g ∆ , and by e ∈ P ∆ the image g ∆ (1 , H ∆ . The linear system | H ∆ | is denoted by L ∆ ,and, given a positive integer m , we let L ∆ ( m ) to be the subsystem of L ∆ consistingof the curves having multiplicity at least m at e . We will denote by π ∆ : X ∆ → P ∆ the blowing up at e ∈ P ∆ and by E the exceptional divisor of π ∆ . Notation 4.1.
Given a triple (∆ , m,
Γ) where ∆ is a lattice polygon, m a positiveinteger and Γ ∈ L ∆ ( m ), the curve Γ is given by a Laurent polynomial (4.1) andthe curve V ( f ) = Γ ∩ G m will also be denoted by Γ. We denote by C the propertransform of Γ in X ∆ . In this section we will investigate properties of pairs ( C, X ∆ ).We drop the subscript ∆ from notation P ∆ , X ∆ , etc. if no confusion arises. Proposition 4.2.
Consider a triple (∆ , m, Γ) as in Notation 4.1. Suppose Γ isirreducible and its Newton polygon is ∆ . The following hold: (i) the arithmetic genus of C is p a ( C ) = 12 (cid:0) Vol(∆) − m + m − | ∂ ∆ ∩ Z | (cid:1) + 1;(ii) any edge F of ∆ of lattice length gives a smooth point p F ∈ C defined asthe intersection of C with the toric boundary divisor corresponding to F .This point is defined over the field of definition of Γ .Proof. Since ∆ is the Newton polygon of Γ, Γ ⊆ P does not contain any torus-invariant point of P . In particular, Γ is contained in the smooth locus of P andhence C is contained in the smooth locus of X . By adjunction formula, p a ( C ) = 12 ( C + C · K X ) + 1 = 12 (Vol(∆) − m + C · K X ) + 1 , where the second equality follows from [CLS11, Prop. 10.5.6]. But C · K X =Γ · K P + m , so that in order to prove (i) we only need to show thatΓ · K P = −| ∂ ∆ ∩ Z | . (4.2)Observe that − K P is the sum of all the prime invariant divisors of P and each primeinvariant divisor D ⊆ P corresponds to an edge F of ∆, see [CLS11, Prop. 10.5.6].Let us fix such an edge F . By a monomial change of variables, we can assume that F lies on the x axis. The inclusion of algebras k [ x , x ± ] → k [ x ± , x ± ] gives theinclusion G m → G m × A , and V ( x ) ⊆ G m × A is an affine open subset of D .Since Γ does not contain any torus-invariant points of P , Γ ∩ D = Γ ∩ V ( x ), andthe latter intersection has equation f | F := (cid:88) u ∈ F ∩ Z α u x u = f (0 , x ) = 0 . (4.3)The degree of this Laurent polynomial is the lattice length of F , so that (4.2) holds.Moreover, if F has length 1, the equation (4.3) has degree 1, which means thatΓ intersects the prime divisor D transversally at a smooth point p F ∈ Γ. Since D isdefined over the base field, if Γ is defined over a subfield k ⊂ k then so is p F . (cid:3) Definition 4.3.
Let ∆ ⊆ R be a lattice polygon with at least 4 vertices. We saythat ∆ is almost good if, for some integer m ,(i) Vol(∆) = m ;(ii) | ∂ ∆ ∩ Z | = m ; iii) dim L ∆ ( m ) = 0, and the only curve Γ ∈ L ∆ ( m ) is irreducible;(iv) the Newton polygon of Γ coincides with ∆;We call the polygon ∆ good if, in addition,(v) The restriction res( C ) = O X ( C ) | C is not torsion. Theorem 4.4. If ∆ is an almost good polygon then ( C, X ∆ ) is an elliptic pair(we call it a toric elliptic pair), e ( C, X ∆ ) > , and C is defined over the base field.If ∆ is good then char k = 0 , e ( C, X ∆ ) = ∞ , and Eff( X ∆ ) is not polyhedral.Proof. Let ∆ be an almost good lattice polygon. The curve Γ is irreducible by (iii)and does not pass through the torus-invariant points of P ∆ by (iv). It follows that C is contained in the smooth locus of X ∆ . Toric surface singularities, i.e. cyclicquotient singularities, are log terminal. By (iv) and [CLS11, Prop. 10.5.6], Γ =Vol(∆), so that (i) is equivalent to C = 0. Finally, conditions (i) and (ii), togetherwith Prop. 4.2, imply that p a ( C ) = 1. Thus ( C, X ∆ ) is an elliptic pair. Observethat O X ( C ) | C = res( C ) ∈ Pic ( C ) (see Definition 3.1), so that condition (v) isequivalent to e ( C, X ∆ ) = ∞ . Suppose this is the case. Since dim L ∆ ( m ) = 0, thecurve Γ, and thus also the curve C , and thus also the line bundle O X ( C ) | C , are alldefined over the base field. In characteristic p , the group (Pic C )( F p ) is torsion,which contradicts e ( C, X ∆ ) = ∞ . Thus char k = 0. Since ∆ has at least 4 vertices, ρ ( X ∆ ) ≥ X ) is not polyhedral by Lemma 3.3. (cid:3) Remark . We don’t know examples of good quadrilaterals.
Example . Polygon 111 is the polygon ∆ with vertices: (cid:20) (cid:21) which appears in Table 3 for m = 7 (where it corresponds to the blue matrix) andwe will use it later in the proof of Theorem 1.3. We claim that ∆ is good.First of all, Vol(∆) = 49 and | ∂ ∆ ∩ Z | = 7 (see Computation 10.1). By Compu-tation 10.2, L ∆ (7) has dimension 0 and the unique curve Γ ∈ L ∆ (7) has equation − u v + 4 u v + 8 u v − u v − u v − u v − u v + 21 u v +6 u v + 40 u v + 85 u v − u v − u v − u v − u v + 56 u v − u v + u + 15 u v + 80 u v − u v + u v + 3 uv − uv +5 uv + 2 uv − v + 4 v = 0 . The exponents of the red monomials are the vertices of ∆, so that the Newtonpolygon of Γ is ∆. By Computation 10.3 the curve Γ is irreducible and its stricttransform C ⊆ X ∆ is a smooth elliptic curve. It has the minimal equation y + xy = x − x − x + 4by Computation 10.6. This is the curve labelled 446.a1 in the LMFDB data-base [LC20]. Since e ( C, X ) >
1, res( C ) ∈ Pic ( C ) is not trivial. Since the Mordell–Weil group is Z , res( C ) is not torsion and therefore ∆ is good. Remark . If in Definition 4.3 we substitute condition (ii) with | ∂ ∆ ∩ Z | < m ,the curve C will have arithmetic genus p a ( C ) >
1, so that (
C, X ∆ ) is no longer anelliptic pair. However, if res( C ) is not torsion, we can still conclude that Eff( X ∆ )is not polyhedral by Proposition 2.3. In the database [Bal20], there are only two olygons satisfying | ∂ ∆ ∩ Z | < m together with (i), (iii) and (iv). Both polygonshave volume 49 and 5 boundary points, so that by Proposition 4.2 the correspondingcurve C has genus 2. In the first case we verified that 2 C moves (Computation 10.2),so res( C ) is torsion. The second polygon has the following vertices (cid:20) (cid:21) and we claim that in this case res( C ) is not torsion. Indeed the curve C is isomorphicto a hyperelliptic curve with equation y + ( x + x + 1) y = x − x + x − x. This is the curve labelled 1415.a.1415.1 in the LMFDB database [LC20] and theMordell-Weil group of the corresponding jacobian surface is isomorphic to Z ⊕ Z / Z .By Computation 10.2, dim | C | = 0 and we conclude that res( C ) is non-torsion.5. Arithmetic toric elliptic pairs of infinite order
Notation 5.1.
Given a lattice polygon ∆ ⊆ Z , let P be the projective toricscheme over Spec Z given by the normal fan of ∆, with a relatively ample invertiblesheaf L given by the polygon ∆. Let X be the blow-up of P along the identitysection of the torus group scheme. Let E (cid:39) P Z be the exceptional divisor. For anyfield k , we denote by P k , L k , X k , E k the corresponding base change (or simply by P , L, X, E if k is clear from the context). We will assume that ∆ is a good polygonin characteristic 0, i.e., ( C C , X C ) is an elliptic pair of order e ( C C , X C ) = ∞ . Then( C C , X C ) is a geometric fiber of an arithmetic elliptic pair ( C , X ) of infinite order flatover an open subset U ⊂ Spec Z , see Definition 3.19. We assume that C C is a smoothelliptic curve . A geometric fiber ( C, X ) of ( C , X ) over a prime p ∈ U is an ellipticpair of finite order e p . There is a morphism of schemes X → Y flat over U inducinga morphism X → Y to the minimal model for any geometric fiber. Geometric fibers( C, Y ) of ( C , Y ) over U are minimal elliptic pairs with Du Val singularities and thesame root lattice T , which we call the root lattice of ∆. Recall that we call p apolyhedral prime of ∆ if Eff( Y ) is a polyhedral cone in characteristic p . We areinterested in the distribution of polyhedral and non-polyhedral primes. Recall thatpolyhedrality is governed by Corollary 3.17: p is polyhedral if and only if thereare roots β , . . . , β − R ∈ E \ ˆ T , linearly independent modulo ˆ T and such thatres( β i ) = 0 in C ( F p ) / res( C ). Here R is the rank of T .We will need a lemma on arithmetic geometry of elliptic curves. Lemma 5.2.
Let C be an elliptic curve defined over Q without complex multiplica-tion over ¯ Q . Fix points x , . . . , x r ∈ C ( Q ) of infinite order. Suppose the subgroup (cid:104) x , . . . , x r (cid:105) ⊂ C ( Q ) generated by x , . . . , x r is free abelian and does not contain amultiple of x . Then reductions ¯ x , . . . , ¯ x r modulo p are not contained in the cyclicsubgroup generated by the reduction ¯ x for a set of primes of positive density.Remark . Note that x , . . . , x r ∈ C ( Q ) are not assumed linearly independent. Proof.
For a fixed integer q , let C [ q ] ⊂ C ( ¯ Q ) be the set of q -torsion points, so that C [ q ] (cid:39) ( Z /q Z ) as a group. Let K be the field Q ( C [ q ]). Since C does not havecomplex multiplication, Gal( K/ Q ) (cid:39) GL ( Z /q Z ) (5.1) or almost all primes q by Serre’s theorem [Ser71]. Choose a basis y , . . . , y s of (cid:104) x , . . . , x r (cid:105) . Since x has infinite order, y = x , y , . . . , y s is a basis of the freeabelian group (cid:104) x , . . . , x r (cid:105) . Choose points y /q, . . . , y s /q ∈ C ( ¯ Q ). Let K y ,...,y s bea field extension of K generated by y /q, . . . , y s /q (any choice of quotients gives thesame field). By Bashmakov’s theorem [Bas72], for almost all primes q we haveGal( K y ,...,y s / Q ) (cid:39) GL ( Z /q Z ) (cid:110) (( Z /q Z ) ) s +1 . For any x ∈ C ( Q ), let i (¯ x ) denote the index of the subgroup (cid:104) ¯ x (cid:105) ⊂ C ( F p ).It suffices to prove that i (¯ x ) , . . . , i (¯ x r ) is not divisible by q but i (¯ x ) is divisibleby q for a set of primes p of positive density. By [LT77], i (¯ x ) is divisible by q if andonly if the Frobenius element σ p = ( γ p , τ p ) ∈ Gal( K x / Q ) (cid:39) GL ( Z /q Z ) (cid:110) ( Z /q Z ) belongs to one of the following conjugacy classes: either γ p = 1 of γ p has eigenvalue 1and τ p ∈ Im( γ p − x i = s (cid:80) j =1 a ij y j for i = 1 , . . . , r , a ij ∈ Z .To apply the Chebotarev density theorem [Tsc26], it remains to note that the subsetof tuples ( γ, τ , . . . τ s ) ∈ GL ( Z /q Z ) (cid:110) (( Z /q Z ) ) s +1 such that γ has eigenvalue 1, τ ∈ Im( γ −
1) and s (cid:80) j =1 a ij τ j (cid:54)∈ Im( γ −
1) for i = 1 , . . . , r , is non-empty for q (cid:29) (cid:3) Remark . We were inspired by the following theorem of Tom Weston [Wes03].Suppose we are given an abelian variety A over a number field F such that End F A is commutative, an element x ∈ A ( F ) and a subgroup Σ ⊂ A ( F ). If red v x ∈ red v Σfor almost all places v of F then x ∈ Σ + A ( F ) tors .Here is another variation on the same theme: Lemma 5.5.
Let C be an elliptic curve defined over Q with points x, y ∈ C ( Q ) ofinfinite order such that y = dx for a square-free integer d . Suppose there exists aprime p of good reduction and coprime to d such that the index of (cid:104) ¯ x (cid:105) is coprimeto d but the index of (cid:104) ¯ y (cid:105) is divisible by d . Then ¯ x, x, . . . , ( d − x (cid:54)∈ (cid:104) ¯ y (cid:105) for a setof primes of positive density.Proof. We need to prove positive density of primes such that the index of thesubgroup (cid:104) ¯ y (cid:105) in (cid:104) ¯ x (cid:105) is equal to d . It is enough to prove positive density for the setof primes such that the index of (cid:104) ¯ x (cid:105) in C ( F p ) is coprime to d but the index of (cid:104) ¯ y (cid:105) isdivisible by d . Arguing as in the proof of Lemma 5.2, we can express this conditionas a condition that the Frobenius element σ p is contained in the union of certainconjugacy classes in the Galois group Gal L/ Q , where L is obtained by adjoiningthe d -torsion C [ d ] and the point x/d . To apply Chebotarev density theorem, weneed to know that this conjugacy class is non-empty. Arguing in reverse, it sufficesto find a specific p such that the condition holds. (cid:3) Theorem 5.6.
Consider good polygons from Table 1 (numbered as in Table 3).We list the root lattice T , the minimal equation of the elliptic curve C , its Mordell-Weil group C ( Q ) and res( C ) . The set of non-polyhedral primes is infinite of positivedensity and includes primes under from Table 2.Proof. We first explain an outline of the argument and then proceed case-by-case.We compute the normal fan of ∆ and the fan of the minimal resolution ˜ P ∆ of P ∆ using Computation 10.1. We use Computation 10.4 to compute the Zariski T C MW res( C )19 A y + y = x − x − x + 54 Z − (1 , A ⊕ A y + y = x + x Z , A ⊕ A y + xy = x − x − x + 4 Z ( − , − A ⊕ A y + y = x + x − x + 1190 Z (15 , Table 1. N primes19 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Table 2. decomposition of K X + C , which by Theorem 3.10 gives curves C , . . . , C s contractedby the morphism to the minimal model Y , and the classes of proper transformsof these curves in ˜ P ∆ . Whenever ∆ has lattice width m in horizontal and verticaldirections, these curves include 1-parameter subgroups C = ( y = 1) and C = ( x =1). We use Computation 10.5 to compute the root lattice T , Cl ( Y ), and the push-forward map to Cl ( Y ). Computation 10.2 gives the equation of the unique memberΓ of the linear system L ∆ ( m ) and its Newton polygon and Computation 10.3 showsthat the proper transform C of this curve in X is an elliptic curve. It follows that∆ is good in characteristic 0. We use Computation 10.6 to compute the minimalequation of C , intersection points of C with the toric boundary divisors, res( C )and the images of roots in E . In the same Computation we apply Corollary 3.17to test polyhedrality of specific primes from Table 2. Finally, we apply Lemma 5.2or Lemma 5.5 to prove positive density of non-polyhedral primes. (cid:3) Example . Polygon 19 has vertices (cid:20) (cid:21) (5.2)The minimal resolution ˜ P ∆ has the fan from Figure 1, where bold arrows indicatethe fan of P ∆ . Note that ˜ P ∆ has a toric map to P × P and proper transformsof 1-parameter subgroups C , C are preimages of rulings. Thus they have self-intersection − e . The minimal resolution of X contains the onfiguration of curves from the right of Figure 1 (toric boundary divisors andcurves C , C ). Only curves C and C contribute to the Zariski decomposition -2 -2 C -1 C -6-2-1-4-2-2-2-2-2 Figure 1.
Polygon 19of K + C and are contracted by the morphism X → Y . Equivalently, the surface Y is obtained by contracting the chain of rational curves above. After blowingdown ( − − Y has an A singularity and Picard number 3. There are two conjugateclasses of root sublattices of type A in E . In our case Cl ( Y ) (cid:39) Z is torsion-free,thus the embedding is primitive. More precisely, we have Cl ( Y ) = E / A , whichcorresponds to the Z -grading of the Lie algebra e = (cid:76) ¯ β ∈ Cl ( Y ) ( e ) ¯ β of the form C Λ C Λ C gl Λ C Λ C C α be a generator of Cl ( Y ). The images of the roots of E are ± kα for k ≤ k res( α ) (cid:54)∈ (cid:104) res( C ) (cid:105) in char p for k = 1 , , α ) and res( C ). The curve Γ has equation f = X Y + 6 X Y − ∗ X Y − X Y − X Y − X Y + X Y + 11 X Y +38 X Y + 26 X Y − X Y − X Y − X Y + 22 X Y + 16 X Y − X Y − X Y + 10 X Y + 15 X Y + 5 XY − XY + 1 = 0 , and passes through e with multiplicity m = 6. When p (cid:54) = 2 , , C has Newtonpolygon (5.2) and is isomorphic to an elliptic curve with the minimal equation y + y = x − x − x + 54 . The curve C is labelled 997.a1 in the LMFDB database [LC20] and its Mordell-Weil group C ( Q ) (cid:39) Z is generated by Q = (1 ,
5) and P = (6 , − C ) = − Q , res( α ) = P − Q , in particular res( C ) is not torsion in characteristic 0and thus ∆ is good. Thus Eff( X ) and Eff( Y ) are not polyhedral in characteristic 0.In characteristic p , k ¯ P is not contained in the cyclic subgroup of C ( F p ) gen-erated by ¯ Q for k = 1 , , atabase [LC20], C has no complex multiplication. To prove positive density ofnon-polyhedral primes, we apply Lemma 5.2 to x = Q and x k = kP for k = 1 , , Remark . Empirically, about 18% of primes are not polyhedral for this polygon.It would be interesting to obtain heuristics for density of non-polyhedral primes.
Remark . Since C contains an irrational 2-torsion point, the Lang–Trotter con-jecture [LT77] predicts that ¯ Q generates C ( F p ) for the set of primes of positivedensity. If true, the Lang–Trotter conjecture implies that Eff Y is polyhedral incharacteristic p for the set of primes of positive density. However, the Lang–Trotterconjecture is only known for curves with complex multiplication [GM90]. Example . Polygon 24 has vertices (cid:20) (cid:21)
The minimal resolution ˜ P ∆ of P ∆ has the fan from the left side of Figure 2, wherebold arrows indicate the fan of P ∆ . As for the Polygon 19, the proper transformsof 1-parameter subgroups C , C in X have self-intersection − Y , which therefore can be obtained by contractingthe configuration of rational curves from the right of Figure 2. It follows that Y -1-2-2-3 -2-2-2-2-2-1 -4-2 Figure 2.
Polygon 24has Picard number 3 and singularities A and A . The curve Γ has a point ofmultiplicity 6 at e and equation f = − Y + 7 XY − X Y − XY + 6 X Y + 2 X Y + 18 XY + 20 X Y − X Y + 10 X Y − X Y − XY − X Y +6 X Y + 5 X Y − X Y + X Y + 5 XY + 3 X Y − X Y − XY , which has a required Newton polygon when p (cid:54) = 2 ,
3. From the Dynking classi-fication it follows that Cl ( Y ) (cid:39) Z . Let α be a generator. The images of rootsof E are equal to ± kα for 0 ≤ k ≤
4. Thus the polyhedrality condition is that res( α ) (cid:54)∈ (cid:104) res( C ) (cid:105) in char p for k = 1 , , ,
4. The minimal equation of the ellipticcurve C is y + y = x + x . It is the curve 43.a1 from the LMFDB database [LC20] of elliptic curves. ItsMordell-Weil group is Z generated by (0 , C ) = Q = 6 (0 ,
0) andres( α ) = P = − (0 , C ) is not torsion and thus Eff Y isnot polyhedral in characteristic 0 by Theorem 3.16. In characteristic p , k ¯ P is notcontained in the cyclic subgroup of C ( F p ) generated by ¯ Q for k = 1 , , , p = 233, when the index of ¯ P is 1 and theindex of ¯ Q is 6. Example . Polygon 111 (discussed in Example 4.6 and followed through incomputations 10.1–10.6). The corresponding curve has the required Newton poly-gon in all characteristics p (cid:54) = 2 , ,
5. The minimal resolution ˜ P ∆ has the fan fromFigure 3, where bold arrows indicate the fan of P ∆ . Note that ˜ P ∆ has a toric mapto P × P and proper transforms of 1-parameter subgroups C , C are preimagesof rulings ; hence, they have self-intersection − e . The Zariskidecomposition of K + C is 2 C + C + C , where C is a curve whose image in P ∆ has multiplicity 3 at e . The Newton polygon of C has vertices (cid:20) (cid:21) and equation X Y − X Y − XY + 5 XY − X + X − X = 0 . On X the curve C is disjoint from C and C . The minimal resolution of X containsthe configuration of curves from the right of Figure 3 (toric boundary divisors andcurves C , C , C ). The curves C , C , C are contracted by the morphism X → Y .Equivalently, the surface Y is obtained by contracting the chain of rational curvesabove. It follows that the root lattice is A ⊕ A and the Picard number of Y is 3.From the Dynkin classification, we have thatCl ( Y ) = E / A ⊕ A (cid:39) Z . Let α be a generator. The images of the roots of E are equal to ± kα , for 0 ≤ k ≤ p the polyhedrality condition is that k res( α ) (cid:54)∈ (cid:104) res( C ) (cid:105) , for k = 1 , , ,
4. To prove that this holds for a set of primes of positive density,we apply Lemma 5.2 to x i = res( iα ), x = res( C ), for i = 1 , , ,
4. We checkthat the conditions in the Lemma are satisfied. Using the minimal equation ofthe curve C (see Example 4.6) and Computation 10.6, we find that res( α ) = P =(0 ,
2) and res( C ) = Q = ( − , − C (labeled 446.a1 in the LMFDBdatabase [LC20]) has no complex multiplication and has Mordell–Weil group Z × Z generated by P and − Q = ( − , P and Q have infinite orderand no multiple of Q is contained in the subgroup generated by P . Example . Polygon 128.
This is a polygon with vertices (cid:20) (cid:21) The 1-parameter subgroups are in this case { u = 1 } and { u = v } . Figure 3.
Polygon 111The minimal resolution ˜ P of P has the fan from the left side of Figure 4, wherebold arrows indicate the fan of P . The proper transforms of 1-parameter subgroups -3-2-3-2 -2-2-2 -1-1D -1 -3-2 -2 D D -1 Α Α Α Figure 4.
Polygon 128 C , C are the only curves contracted by the map to Y , which can be obtained by ontracting the configuration of rational curves from the right of Figure 4, where wealso indicate three boundary divisors, D , D and D . The root lattice is A ⊕ A ,the Picard number of Y is 4. One of the A ’s is indicated with the chain A , A , A of ( − − E contains two lattices A ⊕ A , one primitive and one non-primitive. In our caseCl ( Y ) (cid:39) Z is torsion-free, and therefore we have the primitive one. Next wedescribe the images in Cl ( Y ) of roots in E . In other words, we have a grading ofthe Lie algebra e by the abelian group Cl ( Y ) e = (cid:88) ¯ β ∈ Cl ( Y ) ( e ) ¯ β and we need to describe the subset of non-empty weight spaces B ⊂ Cl ( Y ). A con-venient interpretation of the lattice E is the lattice K ⊥ ⊂ Pic(Bl P ) with thestandard basis h, e , . . . , e . The positive roots are e i − e j for i < j , h − e i − e j − e k ,2 h − e − . . . − ˆ e i − . . . − ˆ e j − . . . − e and 3 h − e − . . . − e i − . . . − e . The primitivesublattice A ⊕ A is generated by simple roots marked black on Figure 5. It follows e -e e -e e -e e -e e -e e -e h-e -e -e Figure 5. A ⊕ A ⊂ E that the Z grading on E is obtained by pairing with fundamental weights h and e + e + e + e that correspond to white vertices of the Dynkin diagram. The Z grading of e has the following non-empty weight spaces (in coordinates given bypairing with h and e + e + e + e , respectively), where we also indicate dimensions.
46 44 1624 616244 It follows that the subset
B ⊂ Cl ( Y ) is given by the ± columns of the matrix1 0 1 1 1 2 2 2 3 30 1 1 2 3 2 3 4 4 5 (5.3)in the basis u, v , where u (resp. v ) is the image of the simple root h − e − e − e (resp. e − e ). Next we compute vectors u and v in Cl ( Y ). By inspecting Figure 4,we see that, in the minimal resolution Z of Y , h − e − e − e corresponds to the( − D − D and e − e to D − D − A − A − A , which has pushforward D − D on X . Next we compute res( C ), res( u ) and res( v ). The equation of Γ is f = X − X Y − X Y + 33 X Y − X Y − XY + 72 X Y − X Y +44 X Y + 31 XY − X Y + 173 X Y − X Y − X Y − Y − XY +37 X Y − X Y − X Y + 23 X Y + 11 X Y + 9 XY − X Y + 65 X Y − X Y + 65 X Y − X Y − X Y + 3 X Y t has a point of multiplicity 7 at e . Its Newton polygon is ∆ for p (cid:54) = 2 , , , C is y + y = x + x − x + 1190 , which is the curve 29157b1 from the LMFDB database of elliptic curves. It hasMordell–Weil group Z generated by A = (3 , − B = ( − ,
34) and C =( − ,
40) We have res( C ) = (15 ,
34) = − A − B + C, res( u ) = (120 , − − A − B − C, res( v ) = ( − , −
50) = A + B + C. We see that res( C ) is not torsion in characteristic 0 and thus Eff Y is not polyhedral.In characteristic p , the condition of polyhedrality is that there exist two linearlyindependent column-vectors of the matrix (5.3) which, when dotted with the rowvector (res( u ) , res( v )) are contained in the cyclic subgroup of C ( F p ) generated byres( C ). This gives the list of non-polyhedral primes in the table. To prove thepositive density, we apply Lemma 5.2 (with r = 10). Remark . In Example 5.12, by Lemma 5.2 , we get positive density not onlyfor the set of non-polyhedral primes but also for the set of primes p such thatthe Halphen pencil | e p C | on Y has only irreducible fibers. For example, res( C )has order 2 in characteristic 23 and none of the elements of B are contained inthe cyclic subgroup of C ( F ) generated by res( C ). It follows that | C | on Y is aHalphen pencil with only irreducible fibers. This property is stronger than non-polyhedrality: in characteristic 13, res( C ) has torsion 5 and res( u + v ) is containedin the cyclic subgroup generated by res( C ) but no other linearly independent vectorin B is. It follows that Eff( Y ) is not polyhedral but the Halphen pencil | C | on Y contains a reducible fiber with two components and no other reducible fibers.6. Infinite sequences of good polygons
An infinite sequence of pentagons.Notation 6.1.
Let k ≥ m = 2 k + 4. Let ∆ be the pentagonwith vertices (0 , m − , m, m − , m ), ( m − , m − G BF AD D D D D Figure 6.
Polygon ∆ for k = 2, m = 8 heorem 6.2. The polygon ∆ is good for every k ≥ . In particular, Eff Bl e P ∆ is not polyhedral. Furthermore, every prime is a polyhedral prime of ∆ . Notation 6.3.
Consider an elliptic curve C ⊆ P with the Weierstrass equation y = x ( x + ax + b ) , where a = − (12 k + 24 k + 11) , b = 4( k + 1) (3 k + 2)(3 k + 4) . Let x = 2( k + 1)(3 k + 2) , x = 2( k + 1)(3 k + 4) . Consider the following points on C in homogeneous coordinates: d = [0 : 1 : 0] , d = [ x : − x : 1] , ˜ d = [ x : x : 1] ,d = [0 : 0 : 1] , d = [ x : x : 1] . Define rational functions on C as follows: f ( x, y ) = x k +1 ( x − y ) x − x , g ( x, y ) = ( x − x )( x k +1 − x k y − x k +10 ) x k ( x − y ) . Lemma 6.4.
The curve C is a smooth elliptic curve defined over Q . The points d , d , ˜ d , d , d are mutually distinct and have the following properties: (i) The given lines intersect C at the following points, with multiplicities: z = 0 : 3 d , x = 0 : d + 2 d ,y = x : d + d + ˜ d , x = x : d + d + ˜ d . In particular, we have equivalences of divisors on C as follows: d ∼ d , d + d ∼ d + d . (ii) The divisors of zeros and poles of the rational functions f and g are ( f ) = (cid:0) ( m − d + d (cid:1) − (cid:0) ( m − d + d (cid:1) , ( g ) = (cid:0) d + γ ) − (cid:0) d + ( m − d + d (cid:1) , where γ is an effective divisor of degree m − disjoint from d , d , d . (iii) The line bundle O ( d − d ) is not a torsion element of Pic ( C ) .Proof. The discriminant equals 16 a ( b − a ) and is non-zero for all integers k ,hence, the curve C is smooth. Part (i) is immediate noticing that x = x + ax + b has solutions x = x and x = x . It follows from (i) that away from the point atinfinity d , the rational function f has zeros at ( m − d + d and a single pole at d , while at d , there is a pole of order ( m − g has poles at d , d , d (of orders 2, ( m −
3) and 1 respectively) and a zero of order at least 2 at d . After achange of variables u = x − x , v = y + x , we see that C has v − (2 k +1) u = 0 tangentline at (0 , w = v − (2 k + 1) u , one can seethat x k +1 − x k y − x k +10 has multiplicity at least 3 at d . Hence, g has multiplicityat least 4 at d . This proves (ii). To prove (iii), choose d as the identity elementof the Mordell-Weil group C ( Q ). By the Mazur’s theorem [Maz77], it suffices toprove that nd (cid:54) = 0 for 1 ≤ n ≤
12. We check this in Computation 10.7. (cid:3) otation 6.5. We label the sides of ∆ as D , D , D , D , D (see Figure 6). Notethat ∆ is inscribed in the square of side m in the first quadrant, with one vertexat (0 ,
0) and sides labeled, starting from the x -axis and going counterclockwise, G, B, A, F . The normal fans of ∆ and the square give rise to toric surfaces P ∆ and P × P . Let S be the toric surface corresponding to the common refinement of thetwo fans and let π : S → P × P , ρ : S → P ∆ be the corresponding toric morphisms. For each of P ∆ , P × P , S , we denote thetorus invariant divisor corresponding to a ray of the fan by the same letter, so B = (1 , , A = (0 , , F = ( − , , G = (0 , − ,D = ( m − , , D = (1 , − , D = (0 , − ,D = ( − ( m − , m − , D = ( − , . On S we have G = D and ρ contracts divisors A, B, F , while π contracts D , D , D , D . Lemma 6.6.
The following equalities of divisors hold on S : π − B = B + ( m − D + D , π − F = F + ( m − D + D ,π − A = A + 2 D + ( m − D + D , π − G = D + 4 D . Proof. If π : Y → X is the weighted blow-up of a toric surface obtained by adding aray generated by a primitive vector f := αe + βe to a smooth cone of the fan of X generated by primitive vectors e , e , then the multiplicity of V ( f ) in π − V ( e ) is α .(Here V ( r ) is the torus invariant divisor corresponding to the ray r ). (cid:3) As is customary, we view a rational function f on a curve C as the map C → P . Proposition 6.7.
Let φ = ( f, g ) : C → P × P be the morphism given by therational functions f , g . Let U be the open torus in P × P , with coordinates ( u, v ) = ([1 , u ] , [1 , v ]) and let Γ := φ ( C ) ∩ U . There are unique morphisms χ : C → P ∆ , ψ : C → S, that commute with φ and π , ρ as defined in Notation 6.5. Then: (1) The map φ is birational onto its image and the equation of Γ in U is ( uv + 2 x k +20 ) (cid:0) u − x k +10 (cid:1) m − − u k +1 ( v + x ) k +2 (cid:0) u − x k +10 (cid:1) k +2 −− u m − ( v + x ) m − (cid:0) uv + u ( x − x ) + 2 x x k +10 (cid:1) = 0 . The Newton polygon of Γ is ∆ and the multiplicity of Γ at the point q with u = 2 x k +10 , v = − x is m . (2) For D i ( i = 1 , , , ) in P ∆ , we have χ − ( D i ) = d i (see Notation 6.3). (3) The induced map χ : C → Bl q P ∆ is an embedding and the linear system L ∆ ( m ) has C as an irreducible member. Via this identification, we have O ( C ) | C ∼ = O C (2 d − d ) . Furthermore, if E is the exceptional divisor in Bl q P ∆ , then χ − ( E ) is acommon fiber of the maps C → P induced by f and g . In particular, χ − ( E ) ∼ ( m − d + d ∼ ( m − d + d ∼ d + γ ∼ d + ( m − d + d . roof. We first prove (1). Set f ( x, y ) = u , g ( x, y ) = v and solve for x, y . Noticingthat uv = x (cid:0) x k +1 − x k y − x k +10 (cid:1) , we obtain after some calculations that x = u ( v + x ) u − x k +10 , y = x k +2 − u ( x − x ) x k +1 . (6.1)In particular, the map φ is birational onto its image. It follows that ( u, v ) ∈ Γmust satisfy the equation obtained by plugging in the above formulas for x, y in theWeierstrass equation of C . After clearing denominators, this equation is (cid:0) x k +2 − u ( x − x ) (cid:1) = x k +2 ( x + ax + bx ) . This equation has a solution x = x , since the point y = x = x lies on C . Moreprecisely, one can factor out ( x − x ), by noticing that( x k +2 ) − x k +3 ( x + ax + b ) = x k +3 ( x − x )( x − x ) . As Γ is irreducible and x is not always equal to x along C , it follows that ( u, v ) ∈ Γmust satisfy the equation u ( x − x ) − ux k +2 = x k +3 ( x − x ) , where x is as in (6.1). Substituting x with the formula in (6.1) and simplifying u ( u is not constant equal to 0 along Γ, otherwise x = 0) it follows ( u, v ) ∈ Γmust satisfy the given equation. Note, the equation is of type ( m, m ) in P × P .Since each of the maps given by the rational functions f and g has degree m and φ is birational onto its image, it follows that the closure of Γ in P × P is a curveof type ( m, m ). In particular, the equation we obtained is irreducible and definesΓ on U . As already noted, the Newton polygon is inscribed in the square withvertices (0 , , m ), ( m, m, m ), the terms u m − v m − , u m − v m , u m v, u m v i except when i = 1,or terms u m − v i when i ≥ k + 3. It follows that the Newton polygon has as anedge the segment joining the points ( m − , m ), ( m, , m − , m −
1) and ( m − , m − m − , m ) respectively, itsuffices to check that there are no terms u i v j with j/i > ( m − / ( m −
3) and noterm u m − v m respectively. This is straightforward. It remains to prove that u m − , u m − , u m − , u m appear with zero coefficients, but u m − has a non-zero coefficient. Setting v = 0,the equation becomes (after simplifying x k +20 )2 (cid:0) u − x k +10 (cid:1) m − − u k +1 (cid:0) u − x k +10 (cid:1) k +2 − x k +10 u m − (cid:0) u ( x − x ) + 2 x x k +10 (cid:1) = 0 . Recall that m = 2 k + 4. Clearly, u m − and u m appear with 0 coefficient. It isstraightforward to check that the coefficients of u m − and u m − respectively, are − x k +10 (cid:18) m − (cid:19) + 4 x k +10 (cid:18) k + 21 (cid:19) − x k +10 ( x − x ) = 0 , and2(2 x k +10 ) (cid:18) m − (cid:19) − x k +10 ) (cid:18) k + 22 (cid:19) − x k +20 x = 0 , respectively . The coefficient of u m − is − x k +10 ) (cid:18) m − (cid:19) + 2(2 x k +10 ) (cid:18) k + 23 (cid:19) , hich is non-zero for all k ≥
0. Making the change of variables s := u − x k +10 , t = v + x , the equation of Γ becomes s m − (cid:0) st + 2 x k +10 t − x s (cid:1) − st ) k +2 (cid:0) s + 2 x k +10 (cid:1) k +1 − t m − (cid:0) st + 2 x k +10 t − x s (cid:1) = 0 , which has a point of multiplicity m at s = t = 0. This finishes the proof of (1).We now prove (2) and (3). Denote d (cid:48) i = χ − ( D i ). Clearly, Vol(∆) = m and | ∂ ∆ ∩ Z | = m . Let C (cid:48) be the proper transform in Bl q P ∆ of the closure of Γ in P ∆ .Note, the map φ factors through C (cid:48) and C is the desingularization of C (cid:48) . Up to achange of coordinates on U , we are in the situation of Proposition 4.2. In particular, C (cid:48) has arithmetic genus one and hence it must be isomorphic to C . We identify C with its image in Bl q P ∆ . As the edges D i for i (cid:54) = 3 of ∆ have lattice length 1, itfollows that each of d (cid:48) i , for i = 1 , , ,
5, is a point. Since C does not pass throughthe torus invariant points of P ∆ , the cycle d (cid:48) is disjoint from d (cid:48) i for i = 1 , , , C embeds into Bl e S and is disjoint from the torus invariant divisors A, B, F .Hence, d (cid:48) i = ψ − ( D i ) for all i . By Lemma 6.6 φ − B = ( m − d (cid:48) + d (cid:48) , φ − F = ( m − d (cid:48) + d (cid:48) ,φ − A = 2 d (cid:48) + ( m − d (cid:48) + d (cid:48) , φ − G = d (cid:48) + 4 d (cid:48) . By the definition of the map φ , the preimages of the torus invariant divisors in P × P are given by the zeros and poles of the rational functions f and g , so byLemma 6.4, these are φ − ( u = ∞ ) = ( m − d + d , φ − ( u = 0) = ( m − d + d ,φ − ( v = ∞ ) = 2 d + ( m − d + d , φ − ( v = 0) = 2 d + γ. where γ is an effective divisor disjoint from d i for i = 1 , , ,
5. By consideringmultiplicities, the only possibility that these divisors match is when d i = d (cid:48) i forall i . For example, the divisor φ − ( v = ∞ ) must equal φ − A , hence, d i = d (cid:48) i for i = 1 , ,
5. Similarly, φ − ( u = ∞ ) must equal one of φ − B or φ − F and as d = d (cid:48) ,it must be that d = d (cid:48) . The exceptional divisor E of Bl q ( P ∆ ) restricts to C as aneffective degree m divisor which is contracted by both maps C → P . Hence, it isa common fiber of the two maps and E | C ∼ ( m − d + d .Up to a change of coordinates on U , the linear system L ∆ ( m ) has C as anirreducible member. To prove that ∆ is a good polygon, it suffices to prove thatres( C ) ∈ Pic ( C ) is non-torsion (this also implies that dim L ∆ ( m ) = 0). Let X = Bl q ( P ∆ ) and let E be the exceptional divisor. We have the following relationsbetween the torus invariant divisors on P ∆ , and hence, on X :( m − D + D ∼ ( m − D + D , D ∼ D − D + ( m − D + D . From the fan of P ∆ we compute D · D = 12(2 m − , D · D = 1 , D · D = 1 m − , D · D = 12 ,D · D = 1 m + 1 , D = 32(2 m − m + 1) , D = − m − m − ,D = − m − m − , D = − m − , D = − ( m − m + 1) , with all other intersection numbers D i · D j being zero. As D , D , D , E are linearlyindependent and generate Cl( X ), from C · D i = 1 ( i (cid:54) = 3), C · E = m we obtain C ∼ m ( m + 1) D + ( m − D − m − D − mE. t follows that res( C ) = 2 d − d . (cid:3) Proof of Theorem 6.2.
By Prop. 6.7(3), up to a change of coordinates on U , thelinear system L ∆ ( m ) has C as an irreducible member. It follows from Prop. 6.7(3)and Lemma 6.4(iii) that res( C ) ∈ Pic ( C ) is non-torsion. This also implies thatdim L ∆ ( m ) = 0 and hence, ∆ is a good polygon. The proper transforms in Bl q S ofthe two one-parameter subgroups C and C of P × P have classes π − B − E and π − A − E , respectively. It follows by Lemma 6.6 that their proper transforms C , C in X = Bl q P ∆ have classes ( m − D + D − E and 2 D + ( m − D + D − E ,respectively. It follows that on X , we have C · C = C · C = 0 and C , C are ( − ρ ( X ) = 4, it follows that the minimal model ( C, Y ) of the ellipticpair (
C, X ) is obtained by contracting C and C and ρ ( Y ) = 2 and every prime ispolyhedral. (cid:3) Remark . The classes of the two one-parameter subgroups C , C can be foundfrom Lemma 6.6. Using the relations between torus invariant divisors, one obtains C ∼ ( m + 1) D + D − D − E, C ∼ ( m − D + D − E. It follows that K + C = ( k + 1) C + ( k + 2) C . For 1 ≤ k ≤ D ⊕ A ⊕ A and the Mordell-Weil groupof C is Z / Z × Z . Remark . The reader may wonder how did we divinate the Weierstrass equationof C in Notation 6.3. We explain how to arrive at the equation of C starting from thepolygon ∆ assuming it can be inscribed in a square with sides of length m . In thiscase, we may add the normal rays of the square to the rays of the normal fan of ∆ toobtain a toric surface S with maps S → P ∆ , S → P × P . If the hypothetical curve C defined by the polygon ∆ is smooth, then the canonical map χ : C → Bl e P ∆ lifts to a map C → Bl e S . The divisor E | C has degree m and is contracted by themaps C → S , and hence also by φ : C → P × P . As the width of ∆ in horizontaland vertical directions is m , the two maps C → P are of degree m . As E | C hasdegree m and is contracted by both maps, it follows that E | C is a common fiber ofboth maps. If φ is given by ( f, g ), where f and g are rational functions on C , itfollows that the divisors of zeros and poles of both f and g (that is, the preimagesof the torus invariant divisors in P × P ) are linearly equivalent to E | C . Thepreimages of the torus invariant divisors of P × P in S can be computed directlyfrom the fan of S (as in Lemma 6.6). Letting d i = χ − ( D i ), where D i are thetorus invariant divisors on P ∆ , we obtain linear relations satisfied by the cycles d i (points if the corresponding edge has lattice length 1) that eventually determinea Weierstrass model of C . For example, for the pentagons in Notation 6.5, oneobtains from Lemma 6.6 and the above argument that E | C ∼ ( m − d + d ∼ ( m − d + d ∼ d + γ ∼ d + ( m − d + d . It follows that 2 d ∼ d , d + d ∼ d + d . Choosing d as the point at infinityand d = (0 ,
0) for an elliptic curve with Weierstrass equation y = x + ax + bx ,we obtain a formula for the rational functions f, g whose zeros and poles are asin Lemma 6.4. Along the way, one has to impose the condition that in the linearsystem given by ( m − d + d ∼ ( m − d + d ∼ d + ( m − d + d thereexists an element vanishing with multiplicity ≥ d . BF AD D D D D D D Figure 7.
Polygon ∆ for k = 2 Remark . Pentagonal curves are fibers C k of an elliptic fibration C → P withthe Weierstrass normal form of Notation 6.3 (the field of rational functions on P isthe field of rational functions in variable k ). By Computation 10.7, C is a rationalelliptic fibration of Kodaira type I I ⊕ I ⊕ . One can compute the Neron–Tateheight of the section of this fibration corresponding to d to conclude that it is nottorsion in the Mordell–Weil group of the elliptic fibration. This shows that d is nottorsion in a fiber C k for almost all k by Silverman’s specialization theorem [Sil83].The Mazur’s theorem gives a more precise statement for every k as above. An infinite sequence of heptagons.
Let k ≥ m = 2 k + 4.Let ∆ be the heptagon with vertices(0 , , (1 , , ( m, , ( m, m − , ( m − , m ) , ( m − , m ) , ( k, k + 1) . Theorem 6.11.
The polygon ∆ is good for every k ≥ . In particular, Eff Bl e P ∆ is not polyhedral in characteristic . Furthermore, for all but finitely many k , theset of non-polyhedral primes of ∆ has positive density.Proof. The strategy is the one in Remark 6.9. The corresponding curve C is asmooth elliptic curve defined over Q with equation y + exy + by = x + ax , where e = − (4 k + 2) , a = − k (2 k + 1) k + 2 , b = 4 k ( k + 1) (2 k + 1)( k + 2) ( k − . Labeling the edges and the corresponding torus invariant divisors in P ∆ as inFigure 7, we let d i = χ − ( D i ). Then d is an effective divisor of degree m − d i for i (cid:54) = 2 are points on C . Let ˜ d be defined by d + ˜ d ∼ d . The points d i have the following properties: d + d ∼ d , d + d ∼ d , d + ˜ d ∼ d . We choose d to be the point at infinity [0 , , C at the following points with multiplicities: x = 0 : d + d + d , y = 0 : 2 d + ˜ d ,x = x : d + d + d , x = − a : d + ˜ d . he points are d = (0 , d = (0 , − b ), ˜ d = ( − a, d = ( − a, ae − b ), d =( x , y ), d = ( x , y ), where y = − y − ex − b and x = 2 k ( k + 1) ( k − k + 2) , y = − k ( k + 1) (5 k + 3)( k − ( k + 2) . The torus invariant divisors D , . . . , D satisfy D + kD +( k +2) D + D ∼ ( m − D + D , D + D +2 D ∼ ( k +1) D +( k +3) D D · D = 1 , D · D = 1 m − , D · D = 12 , D · D = 1 k + 1 ,D · D = 12 , D · D = 1 k + 3 , D · D = 14 ,D = − , D = − − m − , D = − m − , D = kk + 1 − m − ,D = − k + 32( k + 1) , D = − k + 12( k + 3) , D = − k + 114( k + 3) , and the class of C on Bl e P ∆ is given by C ∼ − D + ( m − m − D + mD + mD + ( m + 2) D − mE. There are three one-parameter subgroups C , C , C corresponding to lattice di-rections λ = (1 , λ = (0 , λ = (1 , − m (hence, C · C i = 0 for i = 1 , , Cd + d ∼ d + d ∼ d + ˜ d ∼ d , d + ˜ d ∼ d , O ( C ) | C = O (2 d + 2 d − d ) . There is a map φ : C → P × P , corresponding to rays λ , λ , and given byrational functions f ( x, y ) = x k +1 yα ( x − x −
1) + β ( x + a ) , g ( x, y ) = x + ax k ( x − x ) y , where α = x + a = k (5 k + 3)( k − k + 2) , β = x k +10 y . The pullbacks of the torus invariant divisors of P × P corresponding to the edges A, B, F, G correspond to the zeros and poles of f, g by f = F/B , g = G/A . Themap φ is birational onto its image. Letting u, v be coordinates on P × P andsolving for x, y in f ( x, y ) = u , g ( x, y ) = v , we obtain that φ ( C ) has equation (cid:0) ( αβ ) uv + ( αx ) u − a (cid:1) h ( u, v ) m − − h ( u, v ) m − h ( u, v ) v ++ (cid:0) β ( b + ex ) uv + ( ex α ) u − b (cid:1) h ( u, v ) k h ( u, v ) k +2 h ( u, v ) v = 0 , where h ( u, v ) = ( x β ) uv + ( x α ) u, h ( u, v ) = ( β ) uv − , h ( u, v ) = ( x α ) u + x . It is straightforward to check that this equation has ∆ as its Newton polygon.One computes the classes of the one-parameter subgroups C , C , C as C ∼ ( k + 1) D + ( k + 3) D − E, C ∼ ( m − D + D − E,C ∼ ( m − D + D + D + D − E. It follows that K + C ∼ C + ( k + 1) C + ( k + 1) C .Computation 10.7 (based on Mazur’s theorem as in §
6) shows that O ( C ) | C is nottorsion for k ≥ O (2 d − d − d ) is not torsion. In particular, ∆ is a ood polygon in this range. The point p ∈ C such that O (2 d − d − d ) = O ( p − d )is given by x = 4 k ( k + 1) (2 k + 1)( k − ( k + 2) , y = 4 k ( k + 1) (2 k + 1)(3 k + 1)( k − ( k − . Denote X = Bl e P ∆ and let π : X → Y be the map that contracts the oneparameter subgroups C , C and C . We now compute directly generators Cl( Y )and Cl ( Y ). The group Cl( X ) is generated over Z by D , D , D , D , D and E .An element (cid:88) i ∈{ , , , , } a i D i − bE ∈ Cl( X )belongs to C ⊥ if and only if (cid:80) a i = mb . Hence, C ⊥ is generated over Z by D − D , D − D , D − D , D − D , E − mD . Denote D i , E the classes of D i , E in Cl( Y ). Setting the classes of C , C , C tozero, we obtain the following relations in Cl( Y ): D = 3 D − D , D = 2 D − D , E = ( m + 2) D − D . It follows that Cl( Y ) is generated by D , D , D and the subgroup C ⊥ Y ⊆ Cl( Y ) isgenerated by α := D − D , β := D − D , with D − D = 3 α − β , D − D = 2 α − β , E − mD = ( m + 2) α − β . Sincethe class of C Y can be expressed as 6 α − β , it follows that E /T = Cl ( Y ) = Z { α, β } / Z { α − β } ∼ = Z ⊕ Z / Z . The class of C is divisible by 2. In Cl ( Y ) the class C Y = 3 α − β is the uniquenon-zero torsion element. It follows that there is a commutative diagram E /T = Cl ( Y ) res −−−−→ Pic ( C ) / (cid:104) res( C ) (cid:105) (cid:121) (cid:121) E / ˆ T = Cl ( Y ) / torsion −−−−→ Pic ( C ) / (cid:104) res( C ) (cid:105) (6.2)One can compute directly using a resolution of X that the root lattice is T = A ⊕ A ⊕ A . There is a unique embedding of T in E and it follows that the group E /T isisomorphic to Z { a, b } / Z { a + 2 b } , with the roots in E \ ˆ T having image in E /T belonging to the set {± a, ± b, ± ( a + b ) , ± (2 a + b ) } . It follows that in E / ˆ T ∼ = Z ,we have a = ± α and in E / ˆ T ∼ = Z the images of these roots are the classes of {± α, ± α, ± α } . In order to prove that res( β ) (cid:54) = 0 for any β ∈ E \ ˆ T (for somecharacteristic p ), by (6.2) it suffices to prove that res( α ) is not in the subgroupgenerated by res( C ). To prove that this holds for a set of primes of positivedensity, we apply Lemma 5.2 to x i = res( iα ) = O C ( id − id ) , x = res( 12 C ) = O ( d + d − d )for i = 1 , ,
3. We check that the conditions in the lemma are satisfied. The curve C does not have complex multiplication because its j -invariant − k +1) − k + k ) − k + k ) − k + k ) − k + k )+92160( k + k )+141312 k k k k k k s not an integer (see [Sil94, Thm. II.6.1]). We already proved that x is not torsionin Pic ( C ). To prove that x also has infinite order, it suffices to prove that O ( d − d ) is not torsion, which follows again by Computation 10.7 (based on Mazur’stheorem as in § α ) and res( C ) (equivalently, O C ( d − d ) and O C ( d − d )) are linearly independent for almost all k . UsingSilverman’s specialization theorem [Sil09, App. C, Thm. 20.3] for the ellipticfibration defined by all the heptagonal curves C for k ≥ k , which we do by a computercalculation. (cid:3) Remark . The Mordell-Weil group of C is Z for k = 2 and Z × Z for 3 ≤ k ≤ Remark . “Heptagonal” curves C for k ≥ C k of anelliptic fibration C → P with the Weierstrass normal form of Notation 6.3 (here thefield of rational functions on P is the field of rational functions in parameter k ).By Computation 10.7, C is a K3 elliptic fibration of Kodaira type I ⊕ IV ⊕ .7. A smooth toric elliptic pair
In this section we discuss an example of a minimal elliptic pair (
C, Y ) with Y smooth and such that Pic ( C )( Q ) has rank 9. Let ∆ ⊆ Q be the lattice polytopewhose vertices are the 19 columns of the following matrix: (cid:20) . (cid:21) The polygon ∆ has width m := 30, which is obtained along the directions [1 , , , − m and it has m boundary points. It follows that anycurve in the linear system L ∆ ( m ) has arithmetic genus one. A computer calculationshows that L ∆ ( m ) is zero-dimensional and that its unique element is an irreduciblecurve of geometric genus one whose defining polynomial has Newton polygon ∆.Thus we get an elliptic pair ( C, X ), where X is the blowing-up of the toric surfacedefined by ∆ and C is the strict transform of the unique curve linearly equivalentto the following Weil divisor:[
19 30 12 7 7 1 0 0 1 3 6 16 11 29 48 117 187 72 30 − ]where the first 19 entries are the coordinates of the pullbacks D , . . . D of theprime invariant divisors of the toric variety, while the last coordinate is the coef-ficient of the exceptional divisor E . Observe that X is smooth of Picard rank 18.The linear system | K X + C | contains eight disjoint ( − , , , ,
11 at (1 , C above: −
10 0 0 0 0 0 0 0 0 0 0 0 0 1 2 5 8 3 1 −
11 2 1 1 1 0 0 0 0 0 0 0 0 0 1 3 5 2 1 −
11 1 0 0 0 0 0 0 1 1 1 2 1 2 3 8 13 5 2 −
22 3 1 0 0 0 0 0 1 1 1 2 1 3 5 12 19 7 3 −
33 5 2 1 1 0 0 0 0 0 1 3 2 5 8 20 32 12 5 −
53 5 2 1 1 0 0 0 1 1 1 3 2 5 8 19 31 12 5 −
57 11 4 2 2 0 0 0 0 1 2 6 4 11 18 44 70 27 11 − ach of the divisors D , D and D is a ( − X which has intersectionnumber 1 with C , so that it is disjoint from the curves in | K X + C | . This claim canbe easily proved looking at the primitive generators (cid:37) , . . . , (cid:37) of the normal fanof ∆. For example (cid:37) = [0 , (cid:37) = [ − , (cid:37) = [ − ,
2] show that D is a ( − D · C = D · (19 D + 30 D + 12 D ) gives D · C = 1. Asimilar analysis can be performed for the divisors D and D . As a consequenceeach of the three divisors remains a ( − Y , after contracting the curves in | K X + C | . In particular the linear system | C + D + D + D | defines a rational mapwhich factorizes through Y and there it is defined by | − K Y + D + D + D | . Theimage of Y via this linear system is a smooth cubic surface of P whose equation canbe calculated by determining the unique cubic relation between the elements of abasis of H ( X, C + D + D + D ). A distinguished basis of the latter vector space isgiven by a defining polynomial f for C together with three polynomials f , f , f ,such that { f , f i } is a basis of H ( X, C + D i ). If we denote by ϕ i : X → X i the contraction of D i then C + D i is the pullback of ϕ i ( C ) and thus we have anisomorphism H ( X, C + D i ) (cid:39) H ( X i , ϕ i ( C )). The curve ϕ i ( C ) is defined by amodification ∆ i of the polygon ∆ obtained in the following way: the i − i + 1edges are extended up to their intersection point. The latter is an integer point ifand only if the equation (cid:37) i − + (cid:37) i +1 = (cid:37) i holds, equivalently if D i is a ( − X . We display the construction of the polygon ∆ i in the following picture.The normal fan to ∆ i coincides with that of ∆ at all rays but (cid:37) i . The dimensionof the linear system increases by one because a new monomial, corresponding tothe new point, has been added. A minimal model for the curve C has equation y = x + x − x +8357826814810214400. Ordering the facets of ∆ incounterclockwise way, where the first facet is (0 ,
0) – (3 , { , , , , , , , , , , , , } . For each such indexone can compute the point d i ∈ C ( Q ) cut out by the corresponding toric invariantdivisor D i . This information is then used to compute the images of the 240 rootsand to determine the non-polyhedral primes of X . Using Computation 10.8 wefound 85 non-polyhedral primes in the interval [1 , On the effective cone of M ,n For any toric variety X , we denote by Bl e X the blow-up of X at the identityelement of the torus. Let LM n be the Losev–Manin moduli space [LM00], which isalso a toric variety. Its curious feature, noticed in [CT15], is that LM n is “universal”among all projective toric varieties. Moreover, Bl e LM n is universal among Bl e X .Here we make this philosophical statement very precise: Theorem 8.1.
Let X be a projective toric variety. For any n large enough (see theproof for an effective estimate), there exists a sequence of projective toric varieties LM n = X , . . . , X s = X and rational maps induced by toric rational maps Bl e LM n = Bl e X (cid:57)(cid:57)(cid:75) Bl e X (cid:57)(cid:57)(cid:75) . . . (cid:57)(cid:57)(cid:75) Bl e X s = Bl e X. Every map Bl e X k (cid:57)(cid:57)(cid:75) Bl e X k +1 decomposes as a small Q -factorial modification(SQM) Bl e X k (cid:57)(cid:57)(cid:75) Z k and a surjective morphism Z k → Bl e X k +1 . If the cone Eff(Bl e LM n ) is (rational) polyhedral then Eff(Bl e X ) is also (rational) polyhedral. emark . In [CT15] we used an analogous implication that if Eff(Bl e LM n ) is aMori Dream Space then Eff(Bl e X ) is a Mori Dream Space.The second statement in Thm. 8.1 follows from the first, using Lemma 2.2 andthe fact that if Z (cid:57)(cid:57)(cid:75) Z (cid:48) is an SQM, then we can identify Num ( Z ) R = Num ( Z (cid:48) ) R and Eff( Z ) = Eff( Z (cid:48) ). The proof of the first statement in Thm. 8.1 is based on themain technical result of [CT15], which we give here in a slightly reformulated form: Lemma 8.3 ([CT15, Prop. 3.1]) . Let π : N → N (cid:48) be a surjective map of latticeswith kernel of rank spanned by a vector v ∈ N . Let Γ be a finite set of raysin N R spanned by elements of N , which includes both rays ± R spanned by ± v .Let F (cid:48) ⊂ N (cid:48) R be a complete simplicial fan with rays given by π (Γ) (ignore two zerovectors in the image). Suppose that the corresponding toric variety X (cid:48) is projective(notice that it is also Q -factorial because F (cid:48) is simplicial). Then there exists acomplete simplicial fan F ⊂ N R with rays given by Γ and such that the correspondingtoric variety X is projective. Moreover, there exists a rational map Bl e X (cid:57)(cid:57)(cid:75) Bl e X (cid:48) which decomposes into an SQM Bl e X (cid:57)(cid:57)(cid:75) Z and a surjective morphism Z → Bl e X (cid:48) (of relative dimension ). Corollary 8.4.
Let π : N → N (cid:48) be a surjective map of lattices with kernel spannedby vectors v , . . . , v s ∈ N . Let Γ be a finite set of rays in N R spanned by elementsof N , which includes the rays ± R i spanned by ± v i for i = 1 , . . . , s . Let F (cid:48) ⊂ N (cid:48) R be a complete simplicial fan with rays given by π (Γ) (ignore zero vectors in theimage). Suppose that the corresponding toric variety X (cid:48) is projective (notice that itis also Q -factorial because F (cid:48) is simplicial). Then there exists a complete simplicialfan F ⊂ N R with rays Γ ∪ {± R } ∪ . . . ∪ {± R s } and such that the correspondingtoric variety X is projective. Moreover, there exists a sequence of toric varieties X = X , . . . , X s = X (cid:48) and rational maps induced by toric rational maps Bl e X = Bl e X (cid:57)(cid:57)(cid:75) Bl e X (cid:57)(cid:57)(cid:75) . . . (cid:57)(cid:57)(cid:75) Bl e X s = Bl e X (cid:48) such that every map Bl e X k (cid:57)(cid:57)(cid:75) Bl e X k +1 decomposes as an SQM Bl e X k (cid:57)(cid:57)(cid:75) Z k anda surjective morphism Z k → Bl e X k +1 .Proof. We argue by induction on s , the case s = 1 is Lemma 8.3. We can assume v is a primitive vector. Let N (cid:48)(cid:48) = N/ (cid:104) v (cid:105) . We have a factorization of π into π : N → N (cid:48)(cid:48) and π (cid:48) : N (cid:48)(cid:48) → N (cid:48) . Let Γ (cid:48)(cid:48) be the image under π of Γ (ignore zerovectors in the image). Then we are in the situation of Lemma 8.3. For the map π (cid:48) ,we use the step of the induction. (cid:3) Proof of Theorem 8.1.
We follow the same strategy as [CT15].Applying Q -factorialization, we can assume that X is a Q -factorial toric pro-jective variety of dimension r . The toric data of LM n is as follows. Fix generalvectors e , . . . , e n − ∈ R n − such that e + . . . + e n − = 0. The lattice N is gen-erated by e , . . . , e n − . The rays of the fan of LM n are spanned by the primitivelattice vectors (cid:80) i ∈ I e i , for each subset I of S := { , . . . , n − } with 1 ≤ | I | ≤ n − S = S (cid:97) . . . (cid:97) S r +1 into subsets of equal size m ≥ n = m ( r + 1) + 2). We also fix some indices n i ∈ S i , for i = 1 , . . . , r + 1. Let N (cid:48)(cid:48) ⊂ N be a sublattice spanned by the following ectors: e n i + e j for j ∈ S i \ { n i } , i = 1 , . . . , r + 1 . (8.1)Let N (cid:48) = N/N (cid:48)(cid:48) be the quotient group and let π be the projection map. Then wehave the following:(1) N (cid:48) is a lattice;(2) N (cid:48) is spanned by the vectors π ( e n i ), for i = 1 , . . . , r + 1;(3) π ( e n ) + . . . + π ( e n r +1 ) = 0 is the only linear relation between these vectors.It follows at once that the toric surface with lattice N (cid:48) and rays spanned by π ( e n i )for i = 1 , . . . , r + 1, is a projective space P r . Choose a basis f , . . . , f r for the lattice N (cid:48) so that π ( e n ) = − f , . . . , π ( e n r ) = − f r . Fix one of the indices 1 , . . . , r + 1, westart with r + 1. Choose e = (cid:80) i ∈ I e i such that n , . . . , n r (cid:54)∈ I , | I ∩ S | = k , . . . , | I ∩ S r | = k r and | I | = k + . . . + k r . Then π ( e ) = k f + . . . + k r f r and π ( e + e n r +1 ) = ( k + 1) f + . . . + ( k r + 1) f r . It follows that images of the rays of LM n contain all points with non-zero coordi-nates bounded by m . Repeating this for all r + 1 octants shows that the images ofthe rays of LM n span all lattice points within the region illustrated in Figure 8 for r = 2, which contains all rays of X if m is large enough. To be precise, for each i ∈ { , . . . , r } , in the octant spanned by f . . . , f i − , f i +1 , . . . , f r +1 ( f r +1 := π ( − e n r +1 ) = − f − . . . − f r ) , the region containing all the images of rays of LM n is determined by mf , . . . , mf i − , mf i +1 . . . mf r +1 = − mf − . . . − mf r . It remains to notice (see [OP91]) that there exists a Q -factorial projective toricvariety W with rays given by the images of the rays of LM n and that the toricbirational rational map W (cid:57)(cid:57)(cid:75) X is a composition of birational toric morphismsand toric SQMs. Thus we are done by Corollary 8.4. (cid:3) ( m, m )( − m, − m ) Figure 8.
Corollary 8.5.
Let Y be a projective toric surface with lattice Z and with fanspanned by rays contained in the region of Figure 8 for some m ≥ . If Eff(Bl e Y ) is not (rational) polyhedral then Eff( M , m +2 ) is not (rational) polyhedral. roof. We argue by contradiction. If Eff( M ,n ) is (rational) polyhedral then thepseudo-effective cone Eff(Bl e LM n ) is also (rational) polyhedral by Lemma 2.2 and[CT15, Theorem 1.1]. In this case Eff(Bl e Y ) is (rational) polyhedral by Theo-rem 8.1 (and effective estimates in its proof). (cid:3) Variations in the choice of projections used in the proof of Thm. 8.1 can lead tofurther variations and improvements, such as the following:
Corollary 8.6.
Let Y be a projective toric surface with lattice Z and with fanspanned by rays contained in the polygon with vertices ( l, l ) , ( − , l ) , ( − , , ( − l, , ( − l, − l ) , (1 , − l ) , (1 , − , ( l, − , (8.2) for some l ≥ (see the region in Figure 9 for l = 4 ). If Eff(Bl e Y ) is not (rational)polyhedral then Eff( M , l +5 ) is not (rational) polyhedral. ( l, l )( − l, − l ) ( − , l ) (1 , − l ) ( l, − − l, Figure 9.
Proof.
Similarly, we argue by contradiction. If Eff( M ,n ) is (rational) polyhe-dral then the pseudo-effective cone Eff(Bl e LM n ) is also (rational) polyhedral byLemma 2.2 and [CT15, Theorem 1.1]. In this case Eff(Bl e Y ) is (rational) polyhe-dral using the same idea as in the proof of Theorem 8.1. It suffices to prove thatone can project in such a way that the images of the rays of the fan of LM n arecontained in the polygon given by (8.2).The rays of the fan of LM n are spanned by the primitive lattice vectors (cid:80) i ∈ I e i ,for each subset I of S := { , . . . , n − } with 1 ≤ | I | ≤ n −
3. We partition S = S (cid:97) S (cid:97) S , | S | = | S | = l + 1 , | S | = 1 . We fix some indices n i ∈ S i , for i = 1 , S = { n } . Let N (cid:48)(cid:48) ⊂ N be asublattice spanned by the following vectors: e n i + e j for j ∈ S i \ { n i } , i = 1 , . Let N (cid:48) = N/N (cid:48)(cid:48) be the quotient group and let π be the projection map. Then wehave the following:(1) N (cid:48) is a lattice;(2) N (cid:48) is spanned by the vectors π ( e n i ), for i = 1 , , − ( l − π ( e n ) + − ( l − π ( e n ) + π ( e n ) = 0 is the only linear relationbetween these vectors.Choose a basis f , f for the lattice N (cid:48) given by π ( e n ) = f , π ( e n ) = f . Then π ( e n ) = ( l − f + ( l − f . We calculate the images π ( (cid:80) i ∈ I e i ) of the rays of the fan of LM n . Consider thecase when n , n , n / ∈ I . If | I ∩ S | = i , | I ∩ S | = j , then clearly the images ofsuch rays are given by − if − jf and all values 0 ≤ i, j ≤ l are possible. This givesa square P which in the given basis, has coordinates( − l, − l ) , ( − l, , (0 , − l ) , (0 , . If n ∈ I , n , n / ∈ I , the images π ( (cid:80) i ∈ I e i ) will be contained in the translation of P by f = (1 , n / ∈ I , then π ( (cid:80) i ∈ I e i ) is contained in the union of P with its translates by f = (1 , f = (0 ,
1) and f + f = (1 , Q with sides ( − l, − l ) , ( − l, , (1 , − l ) , (1 , . Finally, if n ∈ I , then π ( (cid:80) i ∈ I e i ) will be contained in the translate Q (cid:48) of Q by f = ( l − , l − Q and Q (cid:48) ,i.e., the polygon given in (8.2). (cid:3) Proof of Theorem 1.3.
It suffices to prove that Eff(Bl e LM ) is not (rational) poly-hedral in characteristic 0 or prime characteristics from the statement of the theo-rem. We do a variation of the projection method in the proof of Thm. 8.1. Weproject the lattice Z of the Losev-Manin space LM (spanned by { e , . . . , e } andsubject to the relation (cid:80) i =1 e i = 0) from the following rays of the fan of LM : e + e + e + e , e + e + e + e , e + e + e + e , e + e and e + e + e . Thesevectors generate the kernel of the map π : Z → Z given by (cid:18) − − − − − (cid:19) . The images of the rays of LM via f are points (both black and white) in Figure 10.By Computation 10.1, the rays of the normal fan of Polygon 111 are precisely thewhite points in Figure 10. By Example 4.6, ∆ is a good polygon. By Theorem 4.4,we conclude that Eff( X ∆ ) is not polyhedral in characteristic 0. In Database 9.2,we collect many more good polygons such that their normal fans (sometimes aftera shear transformation) fit into Figure 10. This shows that Eff( M , ) is not poly-hedral in characteristic p for p < § (cid:3) Databases
Database . We give in Table 3 the list of all good polygons with m ≤
7. It isobtained as follows. We consider all lattice polygons of volume up to 49 (mod-ulo equivalence) appearing in the database [Bal20]. We impose the conditions ofDefinition 4.3 using our Magma package. Computation 10.1 gives (i) and (ii). Com-putations 10.2 and 10.3 give (iii), (iv) and the equation of Γ. This leaves 184 latticepolygons and in all the cases the curve C turns out to be smooth by Computa-tion 10.3. Furthermore, for all but one polygon in this list, we also have that thepoint e is an ordinary multiple point of Γ. The exceptional case is Polygon 23 , igure 10. in which case the tangent cone to the curve Γ at e contains a double line. Thecurve C turns out to be tangent to the exceptional divisor at the correspondingpoint, so that also in this case C is smooth. Therefore, for any polygon in thelist, C is a smooth genus 1 curve and moreover, since ∆ has at least 4 verticesand | ∂ ∆ ∩ Z | = m ≤
7, we also have that at least one edge F of ∆ has latticelength 1. By Proposition 4.2 we conclude that the curve C has a rational point p F that we can chose as the origin, so that in what follows we can treat C as anelliptic curve. This fact allows to check the last condition of the definition of agood polygon, i.e. that O X ( C ) | C = res( C ) is not torsion. Indeed, we can computethe minimal equation of the elliptic curve C using Computation 10.6. We are thenable to compute the order d of the torsion subgroup of the Mordell-Weil group ofthe elliptic curve, and we have that res( C ) is not torsion if and only if res( dC ) isnon-trivial. By Definition-Lemma 3.2 this is equivalent to h ( X, dC ) = 1, and thelatter condition can be checked by Computation 10.2.Another approach is to find a multiple of d using Nagell–Lutz Theorem [ST15]:if p is a prime of good reduction for C , then the specialization map induces an in-jective homomorphism of abelian groups C ( Q ) tors → C ( F p ). Therefore, the torsionorder d of C ( Q ) divides the order of C ( F p ) for any prime p of good reduction, whichis easy to compute from the defining equation of Γ. We then find a multiple of d by taking the greatest common divisor of the orders of C ( F p ) as p varies. (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) Table 3: List of good polygons for m ≤ Database . A database of good polygons used to prove Theorem 1.3 (see theend of § M ,n is not polyhedralfor n ≥
10 in characteristic p for any prime p < vertices non-polyhedral primes39 (cid:3) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (cid:2)
10 10 9 6 3 0 2 74 3 1 0 1 10 9 6 (cid:3) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (cid:2) (cid:3) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (cid:2) (cid:3) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (cid:2) (cid:3) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (cid:2) (cid:3) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (cid:2)
13 9 5 4 2 1 0 1 118 0 3 4 7 9 12 13 9 (cid:3) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (cid:2) (cid:3) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (cid:2) (cid:3) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (cid:2) (cid:3) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (cid:2) (cid:3) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (cid:2) (cid:3) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Table 4:40 Computations
We give an overview of the MAGMA package, which can be downloaded from: https://github.com/alaface/non-polyhedral and contains descriptions of all functions. We first use Polygon 111 as a runningexample, then study infinite families of pentagons and heptagons from Section 6,and finally find non-polyhedral primes up to 2000 for the polygon of Section 7.
Computation . Normal fan of the lattice polygon ∆, the fan of the minimalresolution of the toric surface P ∆ , Vol(∆), number of boundary points. > pol := Polytope([[6,1],[5,4],[1,3],[8,2],[0,6],[0,7],[3,0]]);Transpose(Matrix(Reorder(Rays(NormalFan(pol)))));[ 3 -1 -1 -2 -3 1 3][ 2 3 2 -3 -5 0 1]> Transpose(Matrix(Reorder(Rays(Resolution(NormalFan(pol))))));[ 3 1 0 -1 -1 -1 -1 -1 -2 -3 -1 0 1 3 2][ 2 1 1 3 2 1 0 -1 -3 -5 -2 -1 0 1 1]> [Volume(pol), Computation . Dimension of linear systems L ∆ ( m ) and L k ∆ ( km ) (over differentfields), equation f of Γ ⊂ G m , Newton polytope of f . > m := Width(pol); Computation . Irreducibility and geometric genus of Γ. > IsIrreducible(FindCurve(pol,m,Rationals()));true> Genus(FindCurve(pol,m,Rationals()));1
Computation . In the minimal resolution ˜ X of X , a divisor linearly equivalentto the pullback of C together with the prime components of the pullback of K X + C ,their multiplicities, Newton polygons and equations. > AdjSys(pol);[ [ 19, 7, 2, 1, 0, 0, 0, 0, 0, 1, 2, 5, 8, 20, 13, -7 ],[ 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 2, -1 ],[ 1, 0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 1, 1, 2, 1, -1 ],[ 8, 3, 1, 1, 0, 0, 0, 0, 0, 1, 1, 2, 3, 8, 5, -3 ]]> MultAdjSys(pol);
2, 1, 1 ]> PolsAdjSys(pol);[ x[1] - 1,x[1] - x[2],x[1]^3*x[2] - 3*x[1]^2*x[2] - x[1]*x[2]^2 + 5*x[1]*x[2] - x[1] + x[2]^3 -2*x[2]^2]
Computation . Root lattice of ∆, the map Cl( X ) → Cl( Y ), intersection matri-ces of X and Y (the latter is not in a basis). > RootLat(pol);A6 A1> Cl,g := MapToY(pol);Cl;Full Quotient RSpace of degree 3 over Integer RingColumn moduli:[ 0, 0, 0 ]> imatX(pol);[-10/33 1/11 0 0 0 0 1/3 0][ 1/11 -8/11 1 0 0 0 0 0][ 0 1 -9/7 1/7 0 0 0 0][ 0 0 1/7 -11/7 1 0 0 0][ 0 0 0 1 -3/5 1/5 0 0][ 0 0 0 0 1/5 -12/5 1 0][ 1/3 0 0 0 0 1 -2/3 0][ 0 0 0 0 0 0 0 -1]> imatY(pol);[ 17/14 8/7 13/14 25/14 4/7 31/14 1/2 103/14][ 8/7 3/7 9/7 6/7 5/7 15/7 0 39/7][ 13/14 9/7 5/14 29/14 1/7 27/14 1/2 87/14][ 25/14 6/7 29/14 17/14 10/7 39/14 1/2 135/14][ 4/7 5/7 1/7 10/7 -1/7 11/7 0 23/7][ 31/14 15/7 27/14 39/14 11/7 45/14 3/2 201/14][ 1/2 0 1/2 1/2 0 3/2 -1/2 3/2][103/14 39/7 87/14 135/14 23/7 201/14 3/2 573/14] Computation . Minimal equation of C and images of intersection points withthe toric boundary divisors using the standard MAGMA algorithm, res( C ) andimages of roots in Pic ( C ) (identified with C ), polyhedrality of specific primes. > E,u := EllCur(pol);E;Elliptic Curve defined by y^2 + x*y = x^3 - x^2 - 4*x + 4 over Rational Field> Cl,g := MapToY(pol);C := FindCurve(pol,Width(pol),Rationals());A := Ambient(C);f := Equation(C);h := map
1) we use the following variation: > K
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Universit´e Paris-Saclay, UVSQ, CNRS, Laboratoire de Math´ematiques de Versailles,78000, Versailles, France
E-mail address : [email protected] Departamento de Matem´atica, Universidad de Concepci´on, Casilla 160-C, Concepci´on,Chile
E-mail address : [email protected] Department of Mathematics and Statistics, University of Massachusetts Amherst,710 North Pleasant Street, Amherst, MA 01003, USA and Laboratory of AlgebraicGeometry and its Applications, HSE, Moscow, Russia
E-mail address : [email protected] Dipartimento di Matematica e Informatica, Universit`a degli studi di Palermo, ViaArchirafi 34, 90123 Palermo, Italy
E-mail address : [email protected]@unipa.it