aa r X i v : . [ m a t h . AG ] D ec GENERALIZED GCD FOR TORIC FANO VARIETIES
NATHAN GRIEVE
Abstract.
We study the greatest common divisor problem for torus invariant blowing-up morphisms of nonsingular toric Fano varieties. Our main result applies the theory ofOkounkov bodies together with an arithmetic form of Cartan’s Second Main theorem, whichhas been established by Ru and Vojta. It also builds on Silverman’s geometric concept ofgreatest common divisor. As a special case of our results, we deduce a bound for thegeneralized greatest common divisor of pairs of nonzero algebraic numbers. Introduction
Let k be a number field, X a nonsingular projective variety over k and Y ( X anonsingular codimension r subvariety, r >
2. As observed by Silverman [15], the heightfunction h E ( · ) for E the exceptional divisor of π : X ′ = Bl Y ( X ) → X the blowing-up of X along Y can be interpreted as a generalized logarithmic greatest commondivisor. In particular, it is of interest to obtain bounds for h E ( · ). The purpose of the present note, is to obtain results of this flavour for torus invariantblowing-up morphisms. These results are consequences of the Main Arithmetic GeneralTheorem, [12], together with the theory of polytopes for torus invariant divisors.
Our main result is stated as Theorem 3.1, which should also be viewed within thecontext of our more computational algorithmic point of view which we undertake in Sections4 and 5. Within the context of toric varieties, Theorem 3.1 becomes rather explicit andalgorithmic. Indeed, developing that point of view is the content of Section 5.
As one special case of what we do here, we obtain a result which gives an unconditionalupper bound for the greatest common divisor of pairs of nonzero algebraic numbers. Westate that result as Theorem 6.2. It is a consequence of a more general result (see Theorem3.1) together with Proposition 6.1.
Keywords: Greatest common divisor; Okounkov bodies; Fano Toric Varieties.
Mathematics Subject Classification (2010):
Primary 14G25. Secondary 14C20; 14J45.
The proof of Proposition 6.1 involves calculations with Okounkov bodies; see Sections 4and 5. We believe that these methods are also of an independent interest. Indeed, they maybe seen, for example, as special cases of our related more general results that we obtainedin [6]. They also have an interpretation in terms of certain invariants that arise within thecontext of K-stability. For a more detailed discussion about those topics, we refer to theworks [7], [4], [2] and the references therein.2.
Arithmetic Preliminaries
We fix arithmetic notation and conventions which resemble those of [1]. Let k be anumber field and M k its set of places. We choose absolute values | · | v , for each v ∈ M k , insuch a way that the product formula holds true with multiplicities equal to one. We also let S be a fixed finite set of places. If D is a Cartier divisor on an (irreducible) projective variety X over k then we denotethe local Weil function of D with respect to a place v ∈ M k by λ D ( · , v ) = λ D,v ( · ) . We also denote the logarithmic height function of such a divisor by h D ( · ) = h O X ( D ) ( · ) . Again, we refer to [1] for more details about Weil and height functions.
In our present notation, the greatest common divisor of a pair of nonzero integers0 = α, β ∈ Z is governed by the formulalog gcd( α, β ) := X prime numbers p min { ord p ( α ) , ord p ( β ) } log p [15]. More generally, it makes sense to define the generalized logarithmic greatest commondivisor for pairs of nonzero algebraic numbers α, β ∈ k × . To this end, as in [15, page 337],we put(2.1) h gcd ( α, β ) := X v ∈ M k min { max {− log | α | v , } , max {− log | β | v , }} .In particular, for pairs of nonzero integers 0 = α, β ∈ Z , it follows from (2.1) that h gcd ( α, β ) = log gcd( α, β ). There is an important geometric interpretation of the generalized greatest commondivisor (2.1). This point of view is emphasized by Silverman, for example [15] and [14, page204] (see also [16, Proposition 3] and [17, Proposition 5]). For example, let S (Σ) := P k × P k ENERALIZED TORUS INVARIANT GCD 3 and π : Bl { pt } ( S (Σ)) → S (Σ)the blowing-up of S (Σ) at the origin { pt } ∈ S (Σ) with exceptional divisor E .Fixing local height functions λ E,v ( · ), for each v ∈ M k , it was noted in [15] that if( α, β ) := [1 : α ] × [1 : β ] ∈ S (Σ),with α, β ∈ k × , then h gcd ( α, β ) = X v ∈ M k λ E,v ( π − ( α, β )) + O(1) = h E ( π − (( α, β ))). Our arithmetic results are consequences of the Main Arithmetic General Theorem [12].
Theorem 2.1 ([12]) . Let X be a projective variety over a number field k . Let D , . . . , D q benonzero effective Cartier divisors on X , defined over k . Assume that the divisors D , . . . , D q intersect properly and put D = D + · · · + D q . Let L be a big line bundle on X and definedover k . Let S ⊆ M k be a finite set of places. Then, for each ǫ > , the inequality X v ∈ S λ D ( x, v ) (cid:18) max j q (cid:26) β ( L, D j ) (cid:27) + ǫ (cid:19) h L ( x ) + O(1) holds true for all k -rational points x ∈ X ( k ) outside of some proper Zariski closed subset of X . Here, in Theorem 2.1,(2.2) β ( L, D j ) := Z ∞ Vol( L − tD j )Vol( L ) d t ,for j = 1 , . . . , q , denotes the expected order of vanishing of L along D j . This form of Theorem2.1, stated using the asymptotic volume constants (2.2), has also been noted, for example,in [13, Theorem 1.8] and [7, Theorem 1.1]. Finally, that the divisors D , . . . , D q intersectproperly, means that for each subset I ⊆ { , . . . , q } and all x ∈ T i ∈ I Supp D i , the sequence( f i ) i ∈ I , for f i a local defining equation for D i , is a regular sequence in O X,x , the local ring of X at x .3. The arithmetic general theorem for Fano varieties and consequencesfor greatest common divisors
In this section, we discuss the geometric aspect to the generalized greatest commondivisor problem. What we discuss here is mostly based on [15]. In particular, our mainaim is to derive the inequality (3.10) below. This is an unconditional gcd bound, withmore restrictive assumptions. It may be seen by analogy with that which was observed in[15, Theorem 6]. In what follows, when no confusion is likely, we identify Cartier divisors
NATHAN GRIEVE with linear equivalence classes thereof. On the other hand, note that in applications of theMain Arithmetic General Theorem (Theorem 2.1), for example, it is important to make thedistinction between a given Cartier divisor and its linear equivalence class.
Let X be a nonsingular (irreducible) projective variety, over a number field k , and let Y ( X be a nonsingular codimension r subvariety, r >
2. Let(3.1) π : X ′ = Bl Y ( X ) → X be the blowing-up of X along Y with exceptional divisor E . Then X ′ is nonsingular and,furthermore, the respective canonical divisor classes of X and X ′ are related by the formula(3.2) K X ′ = π ∗ K X + ( r − E. Hencefourth, we impose the following two assumptions(i) there exist nonzero effective Cartier divisors D , . . . , D q on X for which − K X ∼ lin. equiv. D + · · · + D q and which have the property that the divisors π ∗ D , . . . , π ∗ D q intersect properly on X ′ ; and(ii) the divisor − K X is big. In particular, we may assume that − π ∗ K X = π ∗ D + · · · + π ∗ D q ;set(3.3) γ := max j q (cid:26) β ( − K X ′ , π ∗ D j ) (cid:27) .Fix a real number δ >
0, which depends on the given choice of divisors D , . . . , D q , andwhich satisfies the condition that(3.4) γ δ. Let ǫ >
0. By the Main Arithmetic General Theorem [12], which applies becauseof our assumptions (i) and (ii), in particular, because the divisors π ∗ D , . . . , π ∗ D q intersectproperly and because − K X is big, we have that(3.5) − X v ∈ S λ π ∗ K X ( v, x ′ ) ( γ + ǫ ) h − K X ′ ( x ′ ) + O(1) (1 + δ + ǫ ) h − K X ′ ( x ′ ) + O(1)for all k -rational points x ′ ∈ X ′ ( k ) \ Z ′ ( k ) and Z ′ ( X ′ some Zariski closed proper subset.Using the relation (3.2), in the equivalent form − K X ′ = − π ∗ K X − ( r − E , ENERALIZED TORUS INVARIANT GCD 5 we may rewrite the inequality (3.5) as(3.6) − X v ∈ S λ π ∗ K X ( v, x ′ ) + h K X ′ ( x ′ ) ( δ + ǫ ) h − π ∗ K X ( x ′ ) − ( δ + ǫ )( r − h E ( x ′ ) + O(1).By rearranging (3.6), we then obtain that(3.7) − X v ∈ S λ π ∗ K X ( v, x ′ ) + h π ∗ K X ( x ′ ) + (1 + δ + ǫ )( r − h E ( x ′ ) ( δ + ǫ ) h − π ∗ K X ( x ′ ) + O(1).Recall, that the height function h π ∗ K X ( · ) may be expressed in terms of local Weil functions.Precisely(3.8) h π ∗ K X ( · ) = X v ∈ S λ π ∗ K X ( v, · ) + X v ∈ M k \ S λ π ∗ K X ( v, · ) + O(1) . Using this relation (3.8), the inequality (3.7) takes the form(3.9) X v ∈ M k \ S λ π ∗ K X ( v, x ′ ) + (1 + δ + ǫ )( r − h E ( x ′ ) ( δ + ǫ ) h − π ∗ K X ( x ′ ) + O(1) . But now, we may use (3.9) to isolate for h E ( x ′ ). In doing so, we obtain the inequality(3.10) h E ( x ′ ) δ + ǫ )( r − ( δ + ǫ ) h − π ∗ K X ( x ′ ) − X v ∈ M k \ S λ π ∗ K X ( v, x ′ ) + O(1). For later use, we summarize the above discussion in the following way.
Theorem 3.1.
Let π : X ′ → X be the blowing-up morphism of a nonsingular projectivevariety X along a nonsingular codimension r subvariety Y ( X , r > . Let E be theexceptional divisor of π . Assume that:(i) there exist nonzero effective Cartier divisors D , . . . , D q on X which have the twoproperties that:(1) the divisors π ∗ D , . . . , π ∗ D q intersect properly on X ′ ; and(2) the anticanonical class of X is represented by D + · · · + D q ; and(ii) the anticanonical class − K X is big.In this context, set γ := max j q (cid:26) β ( − K X ′ , π ∗ D j ) (cid:27) and fix a real number δ > , as in (3.4) , which depends on the given choice of divisors D , . . . , D q , and which satisfies the condition that γ δ . NATHAN GRIEVE
Let ǫ > . Then, with these assumptions, there exists a proper Zariski closed subset Z ′ ( X ′ which has the property that the inequality (3.11) h E ( x ′ ) δ + ǫ )( r − ( δ + ǫ ) h − π ∗ K X ( x ′ ) − X v ∈ M k \ S λ π ∗ K X ( v, x ′ ) + O(1) is valid for all k -rational points x ′ ∈ X ′ ( k ) \ Z ′ ( k ) . Proof.
By assumption, the two conditions (i) and (ii) stated in § (cid:3) In [15], the logarithmic height function h E ( · ) is called the generalized greatestcommon divisor height . The inequality (3.11) is similar to the main strong upper bound forthe generalized greatest common divisor heights that are made possible by Vojta’s MainConjecture. The above two conditions given in § admissible pairs and a logarithmic formulation of Vojta’s Main Conjecture [10, Section 1.2].They also should be compared with the discussion of [15, Section 4].4. Preliminaries about toric varieties
A main aim of the present article is to obtain an explicit form of Theorem 3.1 for thecase of nonsingular Fano toric varieties. In this section, we fix notations and conventions andrecall relevant facts, about toric varieties, which are especially important for our purposeshere. For more details, we refer to [5] and [3].
For example, such nonsingular toric varieties X = X (Σ), defined over a fixed base field k , are determined by a fan Σ ⊆ N R for N ≃ Z d , where d := dim X . Recall that the conditionthat X is smooth is equivalent to the condition that the minimal generators of each (stronglyconvex rational polyhedral) cone σ ∈ Σ form part of a Z -basis for N. Henceforth, we alsoput M := Hom Z (N , Z ) and M R := Hom R (N R , R ). Let v , . . . , v r be the primitive generators for the one dimensional cones in Σ. Recall,that each v i corresponds to a prime torus invariant divisor D i , for i = 1 , . . . , r . Furthermore,the canonical divisor of X is K X = − r X i =1 D i .In particular, the anticanonical divisor − K X = r X i =1 D i ENERALIZED TORUS INVARIANT GCD 7 is a strict normal crossings (snc) divisor.
In general, fixing a divisor D = P ri =1 a i D i , with a i ∈ R , we obtain its polyhedronP D := { m ∈ M R : h m, v i i > − a i , for i = 1 , . . . , r } .A key point to the theory, is that the lattice points of P D , that is the setP D ∩ M,give a torus invariant basis of H ( X, O X ( D )). We may construct an admissible flag Y • , in the sense of [9, Section 1], that consists oftorus invariant subvarieties of X . The idea is to fix an ordering of the prime torus invariantdivisors in such a way that Y i = D ∩ · · · ∩ D i ,for i d . Indeed, in this way, we obtain a flag Y • : X = Y ⊇ Y ⊇ . . . ⊇ Y d − ⊇ Y d = { pt } ,of irreducible (torus invariant) subvarieties of X , which has the two properties that(i) codim X ( Y i ) = i ; and(ii) each Y i is nonsingular at the point { pt } . Now, following [9, Section 6.1], to relate the Okounkov body of ∆( D ), for D a big divisoron X , to its polytope P D , note that the ray generators v , . . . , v d determine a maximal cone σ , in Σ; they also determine an isomorphism φ R : M R ∼ −→ R d ,which is defined by u ( h u, v i i ) i d . Fixing a big Cartier divisor D on X , we may choose such a torus invariant flag Y • withthe property that the restriction of D to U σ , the affine open subset determined by σ is trivial D | U σ = 0.With these conventions, as noted in [9, Proposition 6.1], the Okounkov body ∆( D ) of D with respect to the flag Y • relates to the polytope P D via the rule∆( D ) = φ R (P D ).Finally for later use, we recall thatVol X ( D ) = d ! Vol R d (P D ). NATHAN GRIEVE Expected order of vanishing along torus invariant divisors
Throughout this section X = X (Σ) denotes a smooth projective toric variety definedover an algebraically closed field k . We also fix E a torus invariant prime divisor over X and(5.1) π : X ′ → X a nonsingular model of X with the property that E ⊆ X ′ . Recall, that such divisors E determine torus invariant valuations on the functionfield k ( X ). Moreover, E may be a prime divisor on X ; in that case, the morphism (5.1),is simply the identity morphism. Finally, note that E may be interpreted as a (prime)birational divisor, in the sense of [12, Section 4], over X . Our main goal is to describe a combinatorial algorithm to compute the asymptoticvolume constant(5.2) β ( L, E ) := Z ∞ Vol( π ∗ L − tE )Vol( L ) d t for L = O X ( D ) a big line bundle on X , determined by a big torus invariant divisor on X ,and E a torus invariant prime divisor over X . We may assume that the morphism (5.1) is obtained as a composition of toric blowing-up morphisms. What is the same, the fan Σ ′ of X ′ is obtained from Σ, the fan of X , by wayof successive star subdivisions. Consider the divisor E over the projective plane P which is obtained byway of blowing-up a torus invariant point. In more detail, the fan Σ that corresponds to P has primitive ray generators v = (1 , , v = (0 ,
1) and v = ( − , − { pt } ∈ S = P be the torus invariant point that corresponds to the maximal cone whichis determined by the primitive ray vectors v and v . The fan Σ ′ which determines π : S ′ = Bl { pt } ( S ) → S ,the monoidal transformation of S with centre { pt } has primitive ray generators v ′ = (0 , − v ′ = (1 , v ′ = (0 ,
1) and v ′ = ( − , − E corresponds to v ′ .Moreover, let D ′ i denote the prime torus invariant divisor that corresponds to v ′ i , for i = 1 , D i , for i = 1 , v , v and v on P are given by π ∗ D = D ′ + E , π ∗ D = D ′ and π ∗ D = D ′ + E . ENERALIZED TORUS INVARIANT GCD 9
We now describe a combinatorial algorithm to compute the asymptotic volume constant(5.3) β ( L, E ) := Z ∞ Vol( π ∗ L − tE )Vol( L ) d t for L = O X ( D ) a big line bundle on X , determined by a big torus invariant divisor on X ,and E a torus invariant prime divisor over X . In particular, our approach here allows forcalculation of the quantities (5.3) for finite sequences of toric blowing-up morphisms overthe (relatively) minimal rational surfaces P , P × P and F r , for r > Let γ eff denote the pseudoeffective threshold of L along Eγ eff ( L, E ) = γ eff := sup { t : π ∗ L − tE is effective } .Then the quantity (5.3) takes the form(5.4) β ( L, E ) := Z γ eff Vol( π ∗ L − tE )Vol( L ) d t . Our approach to determining the quantity (5.4) may be seen as a special case of ourmore general results. The idea is to study the polyhedra P π ∗ L − tE , for 0 t γ eff .To begin with, let v ′ , . . . , v ′ r be the primitive ray vectors for the fan Σ ′ and suppose that E corresponds to v ′ . We may write π ∗ L in the form(5.5) π ∗ L = r X i =1 a i D ′ i + a E. The polyhedra P π ∗ L − tE ⊆ P π ∗ L are then cut out by the system of inequalities, for m ∈ M R , m · v ′ > − a + t ; m · v ′ > − a ; . . . ; and m · v ′ r > − a r . Note that a bound for γ eff may be obtained from the polyhedron P π ∗ L . Furthermore,the expected order of vanishing of L along E , especially the quantity (5.4), is then obtainedvia the knowledge of the volumes of the P π ∗ L − tE . Explicitly, recall that(5.6) Vol X ′ ( π ∗ L − tE ) = d ! Vol R d (P π ∗ L − tE )whereas(5.7) Vol X ( L ) = d ! Vol R d (P π ∗ L ). Combining (5.6) and (5.7), we may rewrite the expected order of vanishing, in other wordsthe quantity (5.4), in the form β ( L, E ) = Z γ eff Vol R d (P π ∗ L − tE )Vol R d (P π ∗ L ) d t . Here we treat the case of a torus invariant point x = { pt } ∈ P and anample line bundle L = O P ( a ), for a >
0. Our conventions about star subdivisions are as in § β ( aπ ∗ D , E ) = β ( aD ′ , E ).The polyhedra P aD is a triangle in the fourth quadrant. This triangle has areaVol R (P aD ) = 12 a . Similarly it follows that Vol R (P aD ′ − tE ) = 12 ( a − t )for 0 t a . In particular, it follows that β x ( L ) = Z a a − t a d t = 23 a ,compare with [11, Section 4]. Next we compute β x ( L ) for a torus invariant point x = { pt } ∈ P × P and L an ample line bundle. Our aim is to study the divisors aπ ∗ D + bπ ∗ D − tE ,for a, b ∈ Z > and t ∈ R > , on π : S ′ = Bl { pt } ( S ) → S = P × P the blowing-up of S at the torus invariant point { pt } ∈ S which corresponds to the maximalcone which is determined by v and v . Here, our conventions for the minimal ray generatorsfor P × P are such that v = ( − , v = (0 , v = (1 ,
0) and v = (0 , − aπ ∗ D + bπ ∗ D − tE = φ − (∆ ( aπ ∗ D + bπ ∗ D − tE )) ,or equivalently their interpretation in terms of Okounkov bodies, to determine the nature ofthe functions f ( t ) := Vol S ′ ( aπ ∗ D + bπ ∗ D − tE )Vol S ′ ( aπ ∗ D + bπ ∗ D ) .In doing so, we recover similar calculations which were obtained in [11, Section 4]. ENERALIZED TORUS INVARIANT GCD 11
Henceforth, we assume that a b . Rewriting, the divisor aπ ∗ D + bπ ∗ D ,in terms of the torus invariant prime divisors on S ′ , we obtain that(5.8) aπ ∗ D + bπ ∗ D = aD ′ + aE + bD ′ . It follows from (5.8) that the inequalities which govern the polyhedron P aD ′ + bD ′ + aE arethen given by the following five inequalities m · v ′ = m + m > − am · v ′ = − m + 0 > m · v ′ = 0 + m > m · v ′ = m + 0 > − am · v ′ = 0 − m > − b .In terms of the divisor aπ ∗ D + bπ ∗ D − tE = aD ′ + bD ′ + ( a − t ) E the inequalities that govern its polyhedronP aπ ∗ D + bπ ∗ D − tE = P aD ′ + bD ′ +( a − t ) E are m · v ′ = m + m > − a + tm · v ′ = − m + 0 > m · v ′ = 0 + m > m · v ′ = m + 0 > − am · v ′ = 0 − m > − b .There are three cases to consider(i) the case that 0 t a ;(ii) the case that a t b ; and(iii) the case that t > b .Henceforth, we also set γ eff = a + b .To begin with, we observe thatVol R (P aπ ∗ D + bπ ∗ D ) = Vol R (P aD + bD ) = ab. Then, by determining the area of the polyhedronsP aπ ∗ D + bπ ∗ D − tE ,in each of the above three cases, we obtain that f ( t ) = − t ab for 0 t a ab (cid:0) ab + a − ta (cid:1) for a t b ( a + b − t ) ab for b t a + b . In particular, we recover the calculations which were obtained in [11, Section 4]. Byintegrating f ( t ) over the interval [0 , a + b ], we determine the asymptotic volume constant β ( aD + bD , E ) := Z ∞ Vol( aπ ∗ D + bπ ∗ D − tE )Vol( aD + bD ) d t = a + b A gcd bound for P × P blown-up along a torus invariant point The aim of this section, is to establish an unconditional bound for the generalizedgreatest common divisor of pairs of nonzero algebraic numbers. In what follows, we continueto adopt the notation of Example 5.12.
Let p and q denote the projections of S = S (Σ) = P × P onto the first and secondfactors respectively. Recall, that the anticanonical divisor class is of type (2 ,
2) on P × P .In particular, by linear equivalence, we may write the anticanonical divisor − K S (Σ) in theform − K S (Σ) ∼ p ∗ H + p ∗ H + q ∗ F + q ∗ F where the divisors p ∗ H , p ∗ H , q ∗ F , q ∗ F have the property that their pullback to S ′ = S (Σ ′ )intersect properly. Thus, using linear equivalence, we may write(6.1) − π ∗ K S (Σ) = H ′ + H ′ + F ′ + F ′ where H ′ i denotes the pullback of the type (1 ,
0) divisors p ∗ H i , for i = 1 ,
2, and where F ′ i denotes, similarly, the pullback of the type (0 ,
1) divisors q ∗ F i . In particular, we may apply Theorem 3.1 with respect to the asymptotic volumeconstants β ( − K S (Σ ′ ) , H ′ i )and β ( − K S (Σ ′ ) , F ′ i ),for i = 1 ,
2. To this end, by linear equivalence, it suffices to determine the nature of the β ( − K S (Σ ′ ) , π ∗ D i ),for i = 1 , , , With this end in mind, here we discuss the polyhedron for the ample divisor − K S ′ = − π ∗ K S − E = D ′ + D ′ + D ′ + D ′ + E . ENERALIZED TORUS INVARIANT GCD 13
Note that by solving for − π ∗ K S = − K S ′ + E ,we obtain − π ∗ K S = − K S ′ + E = D ′ + D ′ + D ′ + D ′ + 2 E = D ′ + ( D ′ + E ) + ( D ′ + E ) + D ′ = π ∗ D + π ∗ D + π ∗ D + π ∗ D . We now determine the asymptotic volume constants β ( − K S ′ , π ∗ D i ) := Z ∞ Vol S ′ ( − K S ′ − tπ ∗ D i )Vol S ′ ( − K S ′ ) d t for i = 1 , , β ( − K S ′ , π ∗ D i ) >
78 ,compare with [12, Corollary 1.11] and [8, Proposition 12].
First, we note that the inequalities which determine P − K S ′ are given by m · v ′ = m + m > − m · v ′ = − m + 0 > − m · v ′ = 0 + m > − m · v ′ = m + 0 > − m · v ′ = 0 − m > − − K S ′ has area equal toVol R (P − K S ′ ) = 1 + 1 + 1 + 12 = 72and, furthermore, that Vol( − K S ′ ) = 2 + 2 + 2 + 1 = 7 . Next, we discuss the divisors − K S ′ − tπ ∗ D i , for i = 1 , , π ∗ D = D ′ + E , π ∗ D = D ′ + E , π ∗ D = D ′ and π ∗ D = D ′ . We observe Proposition 6.1 below.
Proposition 6.1.
Let π : S ′ = Bl { pt } ( S ) → S = P × P be the blowing-up at a torusinvariant point { pt } ∈ S with exceptional divisor E . Let D i , for i = 1 , . . . , , be the prime torus invariant divisors determined by the primitive ray vectors in Σ the fan of S . Then, inthis context, it holds true that β ( − K S ′ , π ∗ D i ) = Z ∞ Vol S ′ ( − K S ′ − tπ ∗ D i )Vol S ′ ( − K S ′ ) d t = 1921 .Proof. By linear equivalence and symmetry, it suffices to consider the case that i = 4 andthen determine the areas of the polyhedra P − K S ′ − tπ ∗ D . With this in mind, we will showthat Vol R (P − K S ′ − tπ ∗ D ) = ( − t for 0 t (2 − t ) + 2 − t for 1 t − K S ′ − tπ ∗ D , we need to determine the area Vol R (P − K S ′ − tπ ∗ D ),for 0 t
2, of the region that is bounded by the five inequalities m + m > − − m + 0 > −
10 + m > − m + 0 > − − m > − t .For 1 t
2, we study the fourth and third quadrants. The areas of the regions that arebounded by these inequalities may be described in the following way • the region in the fourth quadrant that is bounded by these inequalities has area equalto 2 − t ; • the region in the third quadrant that is bounded by these inequalities has area equalto (2 − t ) . Combining, it follows thatVol R (P − K S ′ − tπ ∗ D ) = 12 (2 − t ) + 2 − t ,for 1 t t
1, we note • the region in the first quadrant that is bounded by these inequalities has area equalto 1 − t ; • the region in the second quadrant that is bounded by these inequalities has area equalto 1 − t ; • the region in the third quadrant that is bounded by these inequalities has area equalto ; • the region in the fourth quadrant that is bounded by these inequalities has area equalto 1. ENERALIZED TORUS INVARIANT GCD 15
In sum, it follows that Vol R (P − K S ′ − tπ ∗ D ) = 72 − t ,for 0 t − K S ′ along π ∗ D . To begin with, set f ( t ) = Vol S ′ ( − K S ′ − tπ ∗ D )Vol S ′ ( − K S ′ ) . Then Vol S ′ ( − K S ′ ) = 7and f ( t ) = 27 Vol R (P − K S ′ − tπ ∗ D ) . Thus, it then follows that Z f ( t )d t = 27 (cid:18) (cid:19) = 1921 ,as is desired by Proposition 6.1. (cid:3) Next, we combine Proposition 6.1 and Theorem 3.1 to obtain a gcd type bound forthe case of P × P blown-up along a torus invariant point. The idea is to apply Theorem3.1. In particular, we first note that a consequence of Proposition 6.1 is the fact that β ( − K S (Σ ′ ) , π ∗ D i ) = 1921 ,for i = 1 , . . . ,
4. Next, as mentioned in Subsections 6.2 and 6.3, if we use linear equivalenceto write the anticanonical divisor in the form − K S (Σ) = p ∗ H + p ∗ H + q ∗ F + q ∗ F ,it follows that the divisors H ′ , H ′ , F ′ , F ′ intersect properly. Hencefourth, we put γ := 2119and δ := γ − − Theorem 6.2.
Let k be a number field, M k its set of places and S ⊆ M k a finite subset.Let S (Σ) := P k × P k and π : S (Σ ′ ) → S (Σ) the blowing-up of S (Σ) at a torus fixed point { pt } ∈ S (Σ) with exceptional divisor E . Let ǫ > and put δ = . Then there exists aproper Zariski closed subset Z ′ ( S (Σ ′ ) with the property that (6.2) h E ( x ′ )
11 + δ + ǫ ( δ + ǫ ) h − π ∗ K S (Σ) ( x ′ ) + X v ∈ M k \ S λ π ∗ K S (Σ) ( x ′ ) + O(1) for all k -rational points x ′ ∈ S (Σ ′ )( k ) \ Z ′ ( k ) .Proof. Because of our discussions that precede Theorem 6.2, the hypothesis of Theorem 3.1is satisfied, writing − π ∗ K S (Σ) as in (6.1), and moreover, in (3.11), we may put δ = . Indoing so, we obtain the desired inequality (6.2). (cid:3) This work was conducted while I was a postdoctoral fellow atMichigan State University. It is my pleasure to thank many colleagues for their interest anddiscussions on related topics. Finally, I thank an anonymous referee for carefully readingthis article and for offering several helpful comments and suggestions.
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