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THE SET OF TORIC MINIMAL LOG DISCREPANCIES 6 0 FLORIN AMBRO 0 2 Abstract. We describe the set of minimal log discrepancies of n a toric log varieties, and study its accumulation points. J 1 1 Introduction ] G Minimal log discrepancies are invariants of singularities of log vari- A eties. A log variety (X,B) is a normal variety X endowed with an . h effective Weil R-divisor B, having at most log canonical singularities. t a For any Grothendieck point η ∈ X, the minimal log discrepancy of m (X,B) at η, denoted by a(η;X,B), is a non-negative real number. For [ example, a(η;X,B) = 1 − multη(B) for every codimension one point 2 η ∈ X. For higher codimensional points, minimal log discrepancies can v be computed on a suitable resolution of X. 9 2 LetA ⊂ [0,1]beasetcontaining1andletdbeapositiveinteger. De- 4 note by Mld (A) the set of minimal log discrepancies a(η;X,B), where 8 d 0 η ∈ X is a Grothendieck point of codimension d, and(X,B) is a log va- 5 riety whose minimal log discrepancies in codimension one belong to A. 0 For example, Mld (A) = A. In connection to the termination of a se- / 1 h quence of log flips (see [8, 10]), Shokurov conjectured that if A satisfies t a the ascending chain condition, so does Mld (A). Furthermore, under m d certain assumptions, the accumulation points of Mld (A) should cor- d : v respond to minimal log discrepancies of smaller codimensional points. i X This is known to hold for d = 2 (Shokurov [9], Alexeev [1]) and for r any d in the case of toric varieties without boundary (Borisov [3]). The a purpose of this note is to extend Borisov’s result to the case of toric log varieties. Given the explicit nature of the toric case, we hope this will provide the reader with interesting examples. In order to state the main result, define Mldtor(A) ⊂ Mld (A) as d d above, except that we further require that X is a toric variety and B is torus invariant. Note that Mldtor(A) = A. 1 Theorem 1. The following properties hold for d ≥ 2: 1 Thisworkissupportedbya21stCenturyCOEKyotoMathematicsFellowship, and a JSPS Grant-in-Aid No 17740011. 2 1991MathematicsSubjectClassification. Primary: 14B05. Secondary: 14M25. 1 2 FLORINAMBRO (1) We have 2 ≤ s ≤ d s (cid:12)  (cid:12) (x ,...,x ) ∈ Qs ∩(0,1]s,(a ,...,a ) ∈ As Mldtor(A) = { x a (cid:12) 1 s 1 s , d X i i(cid:12) index(x )|index(x ,...,xˆ,...,x ), ∀1 ≤ i ≤ s  i 1 i s i=1 (cid:12)(cid:12)(cid:12) Psi=1(1+(m−1)xi −⌈mxi⌉)ai ≥ 0 ∀m ∈ Z   where for a r(cid:12)ational point x ∈ Qn, we denote by index(x) the smallest positive integer q such that qx ∈ Zn. (2) If A satisfies the ascending chain condition, then so does Mldtor(A). d (3) Assume that A has no nonzero accumulation points. Then the set of accumulation points of Mldtor(A) is included in d 1 {0}∪ Mldtor({ ;n ≥ 1}·A). [ d′ n 1≤d′≤d−1 Equality holds if d = 2, or if {1;n ≥ 1}·A ⊆ A. n We use the same methods as Borisov [3, 4]. The explicit description in (1) is straightforward, whereas the accumulation behaviour in (2) and (3) relies on a result of Lawrence [6] stating that the set of closed subgroups of a real torus, which do not intersect a given open subset, has finitely many maximal elements with respect to inclusion. Finally, we should point out that Mldtor(A) is strictly smaller than d Mld (A) in general. For example, even the set of accumulation points d of Mld (A) (see Shokurov [9] for an explicit description) is larger than 2 {0}∪{1;n ≥ 1}·A, the set of accumulation points of Mldtor(A). n 2 1. Toric log varieties In this section we recall the definition of minimal log discrepancies and their explicit description in the toric case. The reader may con- sult [2] for more details. A log variety (X,B) consists of an algebraic variety X, defined over analgebraically closed field of characteristic zero, endowed with a finite combination B = b B of Weil prime divisors with real coefficients, i i i P such that K + B is R-Cartier. Here K is the canonical divisor of X X X, computed as the Weil divisor of zeros and poles (ω) of a top X rational form ω ∈ ∧dim(X)Ω1 ; it is uniquely defined up to linear C(X)/C equivalence. The R-Cartier property of K +B means that locally on X X, there exists finitely many non-zero rational functions a ∈ C(X)× α and r ∈ R such that K +B = r (a ). α X α α α Let µ: X′ → X be a proper Pbirational morphism from a normal variety X′ and let E ⊂ X′ be a prime divisor. Let ω be a top rational THE SET OF TORIC MINIMAL LOG DISCREPANCIES 3 form on X, defining KX, and let KX′ be the canonical divisor defined by µ∗ω. The real number a(E;X,B) = 1+multE(KX′ −µ∗(KX +B)) is called the log discrepancy of (X,B) at E. For a Grothendieck point η ∈ X, the minimal log discrepancy of (X,B) at η is defined as a(η;X,B) = inf a(E;X,B), µ(E)=η¯ where the infimum is taken after all prime divisors E on proper birational maps µ: X′ → X. This infimum is either −∞, or a non-negative real number. In the latter case, (X,B) is said to have log canonical singularities at η and the invariant is computed as follows. By Hironaka, there exists a proper birational morphism µ: X′ → X such that X′ is nonsingular, µ−1(η¯) is a divisor on X′, and there exists a simple normal crossings divisor E on X′ which supports both µ−1(η¯) and i i KX′ −µ∗(KX +B). ThePn a(η;X,B) = min a(E ;X,B). i µ(Ei)=η¯ Next we specialize these notions to the toric case. We employ standard terminology on toric varieties, cf. Oda [7]. A toric log variety is a log variety (X,B) such that X is a toric variety and B is torus invariant. Thus there exists a fan∆ in a lattice N such that X = T emb(∆) N and B = b V(e ), where {e } is the set of primitive lattice points i i i i i P on the one-dimensional cones of ∆ and V(e ) ⊂ X is the torus invari- i ant prime Weil divisor corresponding to e . The canonical divisor is i K = −V(e ), and the R-Cartier property of K +B means that X i i X P there exists a function ψ: |∆| → R such that ψ(e ) = 1−b for every i, i i andψ|σ islinear forevery coneσ ∈ ∆. Wemay assumethat (X,B)has log canonical singularities, which is equivalent to ψ ≥ 0 or b ∈ [0,1] i for all i. Let e ∈ Nprim∩|∆| be a non-zero primitive vector. The barycentric subdivision with respect to e defines a subdivision ∆ ≺ ∆ and the ex- e ceptional locus of the birational morphism T emb(∆ ) → T emb(∆) N e N is a prime divisor denoted E . It is easy to see that e a(E ;X,B) = ψ(e). e Due to this property, ψ is called the log discrepancy function of (X,B). Minimal log discrepancies of toric log varieties are computed as follows. These are local invariants, so we only consider affine varieties. Thus ∆ consists of the faces of some strongly convex rational polyhe- dral cone σ ⊂ NR and we denote X = TN emb(σ). Assume first that 0 ∈ X is a torus invariant closed point (it is unique since X is affine). 4 FLORINAMBRO Using the existence of good resolutions in the toric category, it is easy to see that a(0;X,B) = min(ψ| ). N∩relint(σ) For the general case, let η ∈ X be a Grothendieck point. There exists a uniquefaceτ ≺ σ suchthatη ∈ orb(τ). Letcanddbethecodimension of orb(τ) and η in X, respectively. The induced affine toric log variety (X′,B′) = (T emb(τ), mult (B)V(e)) N∩(τ−τ) X V(e) e∈τ(1) has a unique torus invariant closed point 0′, and we obtain a(η;X,B) = mld(0′;X′,B′)+d−c. 2. The set of toric minimal log discrepancies Let A ⊆ [0,1] be a set containing 1. Definition 2.1. For an integer d ≥ 1, let Mldtor(A) be the set of d minimal log discrepancies a(η;X,B), where η ∈ X is a Grothendieck point of codimension d and (X,B) is a toric log variety whose minimal log discrepancies in codimension one belong to A. It is easy to see that Mldtor(A) = A. 1 Definition 2.2. For an integer d ≥ 2, define V (A) to be the set of d pairs (x,a) ∈ (0,1]d×Ad satisfying the following properties: (i) x ∈ Qd. (ii) index(x )|index(x ,...,xˆ,...,x ) for 1 ≤ i ≤ d. i 1 i d (iii) d (1+(m−1)x −⌈mx ⌉)a ≥ 0 for all m ∈ Z. i=1 i i i P For x ∈ Qn, index(x) denotes the smallest positive integer q such that qx ∈ Zn. Note that property (ii) means that (1,0,...,0),...,(0,...,0,1) are primitive vectors in the lattice Zd + Zx. Also, it is enough to verify property (iii) for the finitely many integers 1 ≤ m ≤ index(x)−1. For (x,a) ∈ V (A) we denote d d hx,ai = x a . X i i i=1 Proposition 2.3. For d ≥ 2, we have Mldtor(A) = {hx,ai;(x,a) ∈ V (A)}. d [ s 2≤s≤d THE SET OF TORIC MINIMAL LOG DISCREPANCIES 5 Proof. (1) We first show that the right hand side is included in the left hand side. Fix (x,a) ∈ V (A) for some 2 ≤ s ≤ d. s If s = d, let N = Zd + Zx and let σ be the standard positive cone in Rd, spanned by standard basis e ,...,e of Zd. Let 0 ∈ T emb(σ) 1 d N be the invariant closed point corresponding to σ. Then the affine toric log variety d (T emb(σ), (1−a )V(e )) N X i i i=1 has minimal log discrepancy at 0 equal to hx,ai. Indeed, the log discrepancy function ψ = d a e∗ attains its minimum at x, and i=1 i i ψ(x) = hx,ai. Therefore hx,Pai ∈ Mldtor(A). d Assume now that 2 ≤ s ≤ d−1. Let e ,...,e be the standard basis 1 d of Zd, let e = (d−s)e +e − d e , let v = s x e and let d+1 1 2 i=s+1 i i=1 i i P P N = Zd +Zv. Let σ be the cone in Rd generated by e ,...,e . Set 1 d+1 a = a for s+1 ≤ i ≤ d and a = a . Then i 1 d+1 2 d+1 0 ∈ (T emb(σ), (1−a )V(e )) N X i i i=1 is a d-dimensional germ of a toric log variety with minimal log discrepancy equal to hx,ai. Indeed, note first that the log variety is well defined since a = 2 (d−s)a +a − d a ; thelogdiscrepancy functionisψ = d a e∗. 1 2 i=s+1 1 i=1 i i P P There exists e = d+1y e ∈ N ∩relint(σ) where the log discrepancy i=1 i i P function ψ attains its minimum. We may assume y ∈ [0,1] for every i i. If y ∈/ Z, then y = ··· = y = y , hence e = s y e . d+1 s+1 d d+1 i=1 i i Therefore ψ(e) ≥ ψ(v). If y ∈ Z, then s y e ∈ N ∩Prelint(σ), d+1 i=1 i i hence ψ(e) ≥ ψ( s y e ) ≥ ψ(v). We coPnclude that ψ attains its i=1 i i minimum at v. TPherefore hx,ai = ψ(v) ∈ Mldtor(A). d (2) Let (X,B) be a toric log variety with codimension one log discrepancies in A and let η ∈ X be a Grothendieck point of codimension d. We want to show that a(η;X,B) belongs to the set on the right hand side. There exists a unique cone σ in the fan defining X such that η ∈ orb(σ). Let c be the codimension of orb(σ) in X. Then a(η;X,B) coincides with the minimal log discrepancy of the toric log variety (T emb(σ), mult (B)V(e))×Cd−c. N∩(σ−σ) X V(e) e∈σ(1) in the invariant closed point 0. Therefore we may assume that X is affine and η is a torus invariant closed point 0. 6 FLORINAMBRO We have X = T emb(σ), with dimN = d, B = (1−a )V(e ) N i∈I i i with a ∈ A for every i. The log discrepancy functionPψ ∈ σ∨ of (X,B) i satisfies ψ(e ) = a , and we have i i mld(0;X,B) = min(ψ| ). N∩relint(σ) There exists e ∈ N∩relint(σ) such that mld(0;X,B) = ψ(e). It is easy to see that there exists a subset {1,...,s} ⊆ I, with 2 ≤ s ≤ d, such that e ,...,e are linearly independent and e belongs to the relative 1 s interior of the cone spanned by e ,...,e . Let e = s x e , and 1 s i=1 i i P denote x = (x ,...,x ) ∈ (0,1]d,a = (a ,...,a ) ∈ Ad. It is clear that 1 s 1 s mld(0;X,B) = hx,ai, and we claim that (x,a) ∈ V (A). s Indeed, it is clear that x ∈ Qs. Since e is a primitive lattice point i of N, it is also a primitive in the sublattice s Ze + Ze, which is i=1 i P equivalent to index(x )|index(x ,...,xˆ,...,x ) for every 1 ≤ i ≤ s. i 1 i s Finally, let m ∈ Z. We have s (1+mx −⌈mx ⌉)e ∈ N ∩relint(σ), i=1 i i i hence ψ( s (1 + mx − ⌈mPx ⌉)e ) ≥ ψ(e). Equivalently, s (1 + i=1 i i i i=1 (m−1)xP−⌈mx ⌉)a ≥ 0. Therefore (x,a) ∈ V (A). P (cid:3) i i i s 3. The set V˜ (A) d By Proposition 2.3, the limiting behaviour of toric of minimal log discrepancies is controlled by the limiting behaviour of the sets V (A). d The rationalityproperties (i) and (ii) defining V (A) do not behave well d with respect to limits, and for this reason we enlarge V (A) to a new d set V˜ (A), defined only by property (iii), which turns out to have good d inductive properties and limiting behaviour. Definition 3.1. Let A ⊆ [0,1] be a subset containing 1. Define d V˜ (A) = {(x,a) ∈ (0,1]d×Ad; (1+(m−1)x −⌈mx ⌉)a ≥ 0,∀m ∈ Z}. d X i i i i=1 Equivalently, V˜ (A) is the set of pairs (x,a) ∈ (0,1]d×Ad such that d thegroup Zd+Zx does not intersect theset {y ∈ (0,1]d;hy−x,ai < 0}. As before, we denote hx,ai = d x a . i=1 i i P Lemma 3.2. The following equality holds 1 ˜ V (A) = ((0,1]×{0})∪({ ;n ≥ 1}×A), 1 n where the first term is missing if 0 ∈/ A. In particular, ∞ 1 {hx,ai;(x,a) ∈ V˜ (A)} = ·A. 1 [ n n=1 THE SET OF TORIC MINIMAL LOG DISCREPANCIES 7 Proof. Let x ∈ (0,1] such that 1 + (m − 1)x − ⌈mx⌉ ≥ 0 for every integer m. Equivalently, we have sup(⌈mx⌉−mx) ≤ 1−x. m∈Z Assume by contradiction that x ∈/ Q. Then the set {⌈mx⌉−mx} m≥1 is dense in [0,1] (cf. [5], Chapter IV), hence sup (⌈mx⌉−mx) = 1. m∈Z We obtain 1 ≤ 1−x, hence x = 0. Contradiction. Therefore x = p, for integers 1 ≤ p ≤ q with gcd(p,q) = 1. The q above inequality becomes 1 p 1− = max(⌈mx⌉−mx) ≤ 1− , q m∈Z q hence p = 1. Therefore x = 1. (cid:3) q We will need the following result of Lawrence. Theorem 3.3 ([6]). Let T = Rd/Zd be a real torus. (i) Let U ⊂ T be an open subset. The the set of closed subgroups of T which do not intersect U has only finitely many maximal elements with respect to inclusion. (ii) The set of finite unions of closed subgroups of T satisfies the descending chain condition. Theorem 3.4. Assume that A satisfies the ascending chain condition. Then the set {hx,ai;(x,a) ∈ V˜ (A)} satisfies the ascending chain con- d dition. Proof. Assume first that d = 1. By Lemma 3.2, 1 {hx,ai;(x,a) ∈ V˜ (A)} = { ;n ≥ 1}·A. 1 n Bothsets{1;n ≥ 1},Aarenonnegativeandsatisfy theascending chain n condition, hence their product satisfies the ascending chain condition. Let now d ≥ 2 and assume by induction the result for smaller values ofd. Assume bycontradiction that{(xn,an)} isa sequence inV˜ (A) n≥1 d such that hxn,ani < hxn+1,an+1i for n ≥ 1. Since A satisfies the ascending chain condition, we may assume after passing to a subsequence that an ≥ an+1,∀n ≥ 1,∀1 ≤ i ≤ d. i i Assume first that xn ∈/ (0,1)d for infinitely many n’s. After passing to a subsequence, we may assume xn = 1 for every n. Write xn = (1,x¯n) 1 8 FLORINAMBRO and an = (an,a¯n). Then hx¯n,¯ani < hx¯n+1,a¯n+1i for every n ≥ 1, which 1 ˜ contradicts the acc property of the set {hx¯,a¯i;(x¯,a¯) ∈ V (A)}. d−1 Assume now that xn ∈ (0,1)d for every n. We set Un = {x ∈ (0,1)d;hx−xn,ani < 0} and regard Un as an open subset of the torus Td = Rd/Zd. Let Xn be the union of the subgroups of Td which do not intersect Un. By Theorem 3.3.(i), Xn is a finite union of closed subgroups of Td. It is easy to see that Un ⊆ Un+1, hence Xn ⊇ Xn+1 for n ≥ 1. Since (xn,an) ∈ V˜ (A), we have Un ∩ (Zd + Zxn) = ∅. Therefore d xn ∈ Xn for every n. We have hxn,an+1i ≤ hxn,ani < hxn+1,an+1i. Then xn ∈ Un+1, hence xn ∈/ Xn+1. Therefore Xn % Xn+1 for every n ≥ 1, contradicting Theorem 3.3.(ii). (cid:3) Lemma 3.5. The following properties hold: (1) If A is a closed set, then V˜ (A) is a closed subset of (0,1]d× d Ad. (2) Identify (0,1]s with the face x = ··· = x = 1 of (0,1]d. s+1 d Then V˜ (A)∩(0,1]s = V˜ (A). d s (3) Identify [0,1]s with the face x = ··· = x = 0 of [0,1]d s+1 d and assume that A is a closed set. Then V˜ (A)∩(0,1]s = V˜ (A). d s Proof. (1) Let (x,a) ∈ (0,1]d × Ad such that there exists a sequence {(xn,an)} in V˜ (A) with x = lim xn and a = lim an. Fix n≥1 d n→∞ n→∞ m ∈ Z. By assumption, we have d (1+(m−1)xn −⌈mxn⌉)an ≥ 0,∀n ≥ 1. X i i i i=1 Thereexists apositive integern(m)such that⌈mxn⌉ ≥ ⌈mx ⌉forevery i i 1 ≤ i ≤ d and every n ≥ n(m). Therefore d (1+(m−1)xn −⌈mx ⌉)an ≥ 0,∀n ≥ n(m), X i i i i=1 Letting n converge to infinity, we obtain d (1+(m−1)x −⌈mx ⌉)a ≥ 0. X i i i i=1 THE SET OF TORIC MINIMAL LOG DISCREPANCIES 9 Since m was arbitrary, we conclude that (x,a) ∈ V˜ (A). d (2) This is clear. (3) Assume that we have a sequence {(xn,an)} ⊂ V˜ (A) such that n≥1 d lim xn = (x,0,...,0) ∈ (0,1]s and lim an = (a,a ,...,a ). n→∞ n→∞ s+1 d Let m be a positive integer. Note that for s + 1 ≤ i ≤ d we have mxn ∈ (0,1] for n ≥ n(m), hence i lim(1+(m−1)xn −⌈mxn⌉) = 0 for s+1 ≤ i ≤ d. i i n→∞ Therefore s (1+(m−1)x −⌈mx ⌉)a ≥ 0 for every m ≥ 1. Since i=1 i i i Zs + Zx isPincluded in the closure of Zd +Z x, we obtain s (1 + ≥0 i=1 (m−1)x −⌈mx ⌉)a ≥ 0 for m ≤ −1 as well. Therefore (x,a)P∈ V˜ (A), i i i s proving the direct inclusion. For the converse, just note that (x,a) ∈ V˜ (A) is the limit of the s sequence ((x, 1,..., 1),(a,1,...,1)) ∈ V˜ (A). (cid:3) n n d Definition 3.6. For x ∈ R and m ∈ Z, define x(m) = 1+mx−⌈mx⌉. Note that this operation induces a selfmap of the half-open interval (0,1]. For x ∈ Rd and m ∈ Z, define x(m) ∈ Rd componentwise. Since (0,1]d ∩ (Zd + Zx) = {x(m);m ∈ Z}, we have the equivalent description V˜ (A) = {(x,a) ∈ (0,1]d ×Ad;hx(m) −x,ai ≥ 0,∀m ∈ Z}. d Lemma 3.7. Let x ∈ (0,1]d and let a ∈ Ad such that a > 0 for i 1 ≤ i ≤ d. Then there exists a relatively open neighborhood x ∈ U ⊆ (0,1]d such that if y ∈ U and hy(m) −x,ai ≥ 0 for every m ∈ Z, then hy −x,ai = 0. Proof. (1) Assume first that x ∈ (0,1)d. By Theorem 3.3.(ii), the set of closed subgroups of Rd which contain Zd and do not intersect the nonempty open set {y ∈ (0,1)d;hy −x,ai < 0} has finitely many maximal elements with respect to inclusion, say H ,...,H . 1 l If x ∈ H , then H is a rational affine subspace of Rd in an open 1 1 neighborhood x ∈ U ⊂ (0,1)d. Let v ∈ H −x. Since x ∈ (0,1)d, there 1 1 exists ǫ > 0 such that x+tv ∈ H ∩ (0,1)d for |t| < ǫ. In particular, 1 hx+tv−x,ai ≥ 0,thatisthv,ai ≥ 0for|t| < ǫ. Weinferthathv,ai = 0. ThereforeH ∩U iscontainedin{y ∈ (0,1)d;hy−x,ai = 0}. Ifx ∈/ H , 1 1 1 then U = (0,1)d\H is an open neighborhood of x. 1 1 Repeating this procedure, we obtain a neighborhoodU of x, for each i of the closed subgroup H . The intersection U = U ∩ ···∩ U is the i 1 l desired neighborhood. 10 FLORINAMBRO (2) We may assume after a reordering that x = 1 for 1 ≤ i ≤ s and i x ∈ (0,1) for s < i ≤ n. If s = n, we may take U = (0,1]d. Assume i now that s < n. By [5], Chapter IV, there exists a negative integer m 0 such that s hx(m0) −x,ai < mina . i i=1 Let y ∈ (0,1]s × d (⌈m0xi⌉, ⌈m0xi⌉−1) such that hy(m) − x,ai ≥ 0 Qi=s+1 m0 m0 for every m ∈ Z. We claim that y = ··· = y = 1. Indeed, assume 1 s by contradiction that y < 1 for some 1 ≤ j ≤ s. A straightforward j computation gives d hy(m0)−x,ai−m hy−x,ai = hx(m0)−x,ai+ (⌈m x ⌉−⌈m y ⌉)a . 0 X 0 i 0 i i i=1 By the choice of y, we obtain d s (⌈m x ⌉−⌈m y ⌉)a = (m −⌈m y ⌉)a ≤ −a , X 0 i 0 i i X 0 0 i i j i=1 i=1 hence 0 ≤ hx(m0) −x,ai−a . This contradicts our choice of m . j 0 Let x¯ = (x ,...,x ),y¯ = (y ,...,y ),a¯ = (a ,...,a ). We s+1 d s+1 d s+1 d have (x¯,a¯) ∈ V˜ (A) and hy¯(m) − x¯,¯ai ≥ 0 for every m ∈ Z. From d−s Step 1, there exists an open neighborhood x¯ ∈ U¯ ⊂ (0,1)s such that if y¯∈ U¯ then hy¯−x¯,a¯i = 0. Then d ⌈m x ⌉ ⌈m x ⌉−1 U = (0,1]s ×(U¯ ∩ ( 0 i , 0 i )). Y m m 0 0 i=s+1 (cid:3) satisfies the required properties. Lemma 3.8. The following equality holds for d ≥ 1 and a ∈ Ad: {hx,ai;(x,a) ∈ V˜ (A),x ∈ Qd} = {hx,ai;(x,a) ∈ V˜ (A)}. d d Proof. Let (x,a) ∈ V˜ (A). We have x ,...,x < 1 and x = ... = d 1 s s+1 x = 1, where 0 ≤ s ≤ d. If s = 0, then x ∈ Qd and we are done. d Assume s ≥ 1 and set x¯ = (x ,...,x ) and a¯ = (a ,...,a ). Then 1 s 1 s (x¯,a¯) ∈ V˜ (A). Since x¯ ∈ (0,1)s, and there exists a closed subgroup s Zs ⊆ H¯ ⊆ Rs such that x¯ ∈ H¯ ∩ U ⊂ {z¯;hz¯− x¯,a¯i = 0}, by Step x¯ 1 of the proof of Lemma 3.7. Since H¯ is rational, there exists z¯ ∈ Qs∩H∩U . Set x′ = (z¯,1,...,1). Then(x′,a) ∈ V˜ (A), hx,ai = hx′,ai x¯ d and x′ ∈ Qd. (cid:3)