ON ARITHMETIC GENERAL THEOREMS FOR POLARIZED VARIETIES

Transcription

1 ON ARITHMETIC GENERAL THEOREMS FOR POLARIZED VARIETIES NATHAN GRIEVE Abstract. We apply Schmidt s Subspace Theem to establish Arithmetic General Theems f projective varieties over number and function fields. Our first result extends an analogous result of M. Ru and P. Vojta. One aspect to its proof makes use of a filtration construction which appears in wk of Autissier. Further, we consider wk of M. Ru and J. T.-Y. Wang, which pertains to an extension of K. F. Roth s theem f projective varieties in the sense of D. McKinnon and M. Roth. Motivated by these wks, we establish our second Arithmetic General Theem, namely a fm of Roth s theem f exceptional diviss. Finally, we observe that our results give, within the context of Fano varieties, a sufficient condition f validity of the main inequalities predicted by Vojta. 1. Introduction 1.1. The classical they of Weierstrass points of algebraic curves highlights beautiful interplay between the geometry, combinatics, analysis and arithmetic of a given compact Riemann surface and its Jacobian. That sty also suggests that divisial filtrations of linear series on higher dimensional projective varieties are also of independent interest. F example, they should have applications to the arithmetic, moduli and stability of projective varieties In [16], we obtained results in this direction. Indeed, motivated by [23] and [15], we showed how a fm of Roth s theem, f polarized projective varieties, is related to imptant combinatial and moduli theetic invariants. One feature of our [16], is that we expound upon nontrivial connections between: (i) Geometric Invariant They, especially the they of Chow fms of polarized varieties; (ii) the theies of Okounkov bodies and graded linear series; and (iii) Diophantine Geometry. Among other wks, some me recent starting points f our [16] include [4], [6] and [20] The interplay between Diophantine Geometry, Geometric Invariant They and Probability They is well known. In this spirit, some representative wks include [11], [12], [13] and [14]. At the same time, we have made progress in our understanding of concepts within Tic Geometry, f example the theies of test configurations, K-stability and the Mathematics Subject Classification (2010): 14G05, 14G40, 11G50, 11J97, 14C20. 1

2 2 NATHAN GRIEVE Duistermaat-Heckman measures. As two imptant wks in this area, we mention [10] and [5] Returning to questions in arithmetic, an imptant insight of P. Cvaja and U. Zannier was the fact that Schmidt s Subspace Theem, combined with a detailed understanding of ders of vanishing of sections at rational points, could be used to prove the celebrated Siegel s theem on S-integral points of affine curves, [7]. Both Schmidt s Subspace Theem and the filtration construction of Cvaja-Zannier have been further refined and applied on a number of different occasions ([9], [8], [21], [2] and [22]) Our main purpose here is to study these above mentioned topics within the context of wk of Ru-Vojta, [26], which extends earlier wk of Ru, [25]. In doing so, we obtain results which concern the arithmetic of filtered linear series on projective varieties. Our results complement those of [26] and [25]. F example, they pertain to the nature of Weil functions evaluated at rational points outside of proper Zariski closed subsets. In proving our main result, Theem 1.1, we study the main Arithmetic General Theem of [26]. We generalize that result, see Theem 4.1, and give applications, Collary 1.2 and Collary F meprecise statements, animptant aspect of [26] and[25] is the use of Schmidt s Subspace Theem to deduce Arithmetic General Theems. Those results apply to polarized varieties over number fields. Here we view these theems from a broader perspective. Specifically, we use the techniques of [26], which build on those of [21], [2] and [25], to establish Theem 1.1 below. One feature of our results here is that they apply to polarized varieties over number and characteristic zero function fields. Again, as in [26], one key point is an application of Schmidt s Subspace Theem. Although our proof of Theem 1.1 is based on that of [26], here we offer a slight conceptual improvement. Indeed, we express some aspects of the proof using expectations of discrete measures The use of such measures in the study of diophantine and arithmetic aspects of linear series on projective varieties is well established in the literature, [12], [13], [14], [4], [16]. Similar kinds of expectations and discrete measures also play an imptant role in the K- stability of projective varieties, [5]. They also have imptant connections to measures of asymptotic growth of linear series, [6]. As it turns out, the study of those wks was one impetus to the Arithmetic General Theems that we establish here In a related direction, we establish a fm of Roth s Theem f exceptional diviss, see Collary 1.2. That result is motivated by the main theem obtained by Ru-Wang, [27], themainresults ofmckinnon-roth, [23], [24], aswell asourrelatedwks, [15], [16]. Wealso deduce further consequences, Collary 1.3 and Theem 5.1, that pertain to Vojta s main conjecture. As one other complementary result, a fm of Schmidt s Subspace Theem, which involves Weil functions and Seshadri constants f subvarieties, has been obtained by Heier-Levin, [17].

3 ON ARITHMETIC GENERAL THEOREMS FOR POLARIZED VARIETIES To state Theem 1.1, let K be a number field a characteristic zero function field with fixed algebraic closure K. Let F/K be a finite extension, K F K, let S be a fixed finite set of places of K and extend each v S to a place v of F. Suppose that X is a geometrically irreducible, geometrically nmal projective variety over K. Fix nonzero effective Cartier diviss D 1,...,D q on X, defined over F, and fix Weil functions λ Di,v( ) f each D i and each v S. Our conventions are such that each of the Weil functions λ Di,v( ) are nmalized relative to the base field K Let L be a big line bundle on X, defined over K. We consider the asymptotic volume constants of L along each of the Cartier diviss D i : (1.1) β(l,d i ) := Vol(L td i ) dt. 0 Vol(L) Put D = D D q and, f each v S, choose λ D,v ( ), a Weil function f D with respect to v. The proximity function of D with respect to S is then: m S (,D) = λ D,v ( ) Our main result gives a unified picture of the related wks, [23], [15], [26], [27] and [16]. It also generalizes the Arithmetic General Theem of [26]. F example, Theem 1.1 below also applies to the case that K is a characteristic zero function field. It also allows f each of the Cartier diviss D i to be defined over F/K a finite extension of the base field. Theem 1.1. Let L be a big line bundle on X and suppose that the Cartier diviss D 1,...,D q intersect properly. Let ǫ > 0. There exists constants a ǫ,b ǫ > 0 with the property that either: h L (x) a ǫ ; ( ) m S (x,d) max β(l,d j) 1 +ǫ h L (x)+b ǫ 1 j q f all K-rational points x X(K) outside of some proper Zariski closed subset Z X By analogy with [5, Section 3], the constants β(l,d i ) can be identified with the limit expectation of the Duistermaat-Heckman measures. Further, when L is assumed to be very ample, β(l,d i ) is identified with the nmalized Chow weight of L along D i, [16]. We discuss these matters in Section Theem 1.1 implies a fm of Roth s theem f exceptional diviss. Let π: X XF

4 4 NATHAN GRIEVE be the blowing-up of X along a proper subscheme Y X, defined over F, with exceptional divis E. The asymptotic volume constant takes the fm: β(π L,E) = β E (L) = E(ν) = lim m E(ν m). Here E(ν m ) is the expectation of the discrete measures ν m determined by the filtration of H 0 (X F,mL F ) which is induced by F R(L F ), the filtration of the section ring R(L F ) which is determined by E. Again, by analogy with [5, Section 3], the limit expectation E(ν) can also be identified with the limit expectation of the Duistermaat-Heckman measures, compare also with [4, Section 1] and [6, Section 1]. Collary 1.2. If ǫ > 0, then there exists positive constants a ǫ and b ǫ so that either: λ E,v (x) h L (x) a ǫ ; ( ) 1 β E (L) +ǫ h L (x)+b ǫ f all K-rational points x X(K) outside of some proper subvariety Z X. We prove Collary 1.2 in Section One additional implication of Theem 1.1 and Theem 4.1 applies to Fano varieties. F example, we obtain a sufficient condition f validity of the inequalities predicted by Vojta s main conjecture, [29]. This observation, Collary 1.3 below, is a natural extension to [23, Theem 10.1]. It has also been noted by Ru and Vojta, [26, Collary 1.12]. Here, we view it as a consequence of a me general statement; see Theem 5.1. Collary 1.3. Suppose X that is nonsingular, that K X is ample and that D = D D q is a simple nmal crossings divis on X. In this context, if β( K X,D i ) 1, f all i, then Vojta s inequalities hold true. Precisely, if M is a big divis on X, then f all ǫ > 0, there exists constants a ǫ,b ǫ > 0, so that either h KX (x) a ǫ m S (x,d)+h KX (x) ǫh M (x)+b ǫ f all K-rational points x X(K) outside of some proper subvariety Z X We prove Collary 1.3 in Section 5. When K X is very ample, then, as is a consequence of [16, Theem 5.2], the condition that β( K X,D i ) 1 can be expressed in terms

5 ON ARITHMETIC GENERAL THEOREMS FOR POLARIZED VARIETIES 5 of a nmalized Chow weight f the anti-canonical embedding. From this point of view, Vojta s inequalities are implied by a st of Chow stability condition To put matters into perspective, Collary 1.3 raises the question as to the extent to which its hypothesis can be satisfied in nontrivial situations. To that end, Collary 1.3 is a special case of Theem 5.1. That result can be applied in a variety of contexts including the examples mentioned in [23, Section 10]. Studying the hypothesis of Collary 1.3 and Theem 5.1 in detail is a topic, of independent interest, which we do not pursue here In proving Theem 1.1, we establish a fm of Schmidt s Subspace Theem f effective linear series on projective varieties (Proposition 2.1). In Theem 4.1, we also extend the main Arithmetic General Theem of Ru and Vojta, [26]. Finally, we use results fromour[16], whichbuildonresultsandideasfrom[4],[6]and[5], tointerprettheasymptotic volume constant in terms of ders of vanishing along subvarieties. This theme appears in various contexts within Diophantine Approximation, f example [2], [26] and [27]. We make those connections precise in Section Notations and conventions. Throughout this article, K denotes a number field a characteristic zero function field. Further, F/K denotes a fixed finite extension of K contained in K a fixed algebraic closure of K. All absolute values, height functions and Weil functions are nmalized relative to K. By a variety, we mean a reduced projective scheme over a given base field. If L is a line bundle onageometrically irreducible variety X, defined over K, thenl F denotes itspullback to X F := X Spec(K) Spec(F). Using terminology that is consistent with [26], we say that a collection of Cartier diviss D 1,...,D q on X, defined over F, intersect properly if f all nonempty subsets I {1,...,q} and each x i I D i, the collection of local equations f the D i near x fm a regular sequence in the local ring O X,x. We also denote the greatest lower bound of D 1,...,D q by i I D i and their least upper bound by i I D i. In particular, in line with [26], we may realize these (birational) diviss as Cartier diviss on some nmal proper model of X. Finally, when no confusion is likely, we often times use additive (as opposed to multiplicative) notation to denote tens products of coherent sheaves Acknowledgements. I thank Aaron Levin and Steven Lu f helpful discussions and encouragement. I also thank Julie Wang f conversations and comments on related topics and f sharing with me a preliminary version of her joint wk [27]. Finally, I

6 6 NATHAN GRIEVE thank anonymous referees f their remarks and am especially grateful to Min Ru f helpful comments and f sharing with me the final version of his joint wk [26]. This wk was conducted while I was a postdoctal fellow at Michigan State University. 2. Preliminaries In this section, we make precise concepts and conventions which are required f what follows. We also establish Proposition 2.1 which we use, in Section 4, to establish Theem 4.1. Our approach to absolute values and Weil functions is similar to [15, Section 2] and [3, Section 1] Absolute values and the product fmula. Let K be a number field a characteristic zero function field with set of places M K. F each v M K, there exists a representative v so that the collection of such absolute values v, v M K, satisfy the product rule: v M K α v = 1, f each α K. Our choice of representatives v, v M K, are made precise in the following way. When K = Q, if p M Q and p =, then p is the dinary absolute value on Q. If p M Q is a prime number, then p is the usual p-adic absolute value on Q, nmalized by the condition that p p = p 1. F a general number field K, if p M K, then p p f some p M Q and we put: p := NKp/Q p ( ) 1/[K:Q] p. Here N Kp/Qp ( ) : K p Q p is the nm from K p to Q p. When K is a function field, K = k(y) f (Y,L) a polarized variety over an algebraically closed base field k, char(k) = 0, and Y is irreducible and nonsingular in codimension one. The local ring O Y,p of a prime divis p Y is a discrete valuation ring with discrete valuation d p ( ); put: p := e dp( )deg L (p). In this context, we also write M K = M (Y,L) to indicate dependence on the polarized variety (Y,L). In general, given a finite extension F/K, if w M F is a place of F lying over a place v M K, then we put: Further, let: w := NFw/K v ( ) v. w,k := NFw/K v ( ) [Fw:Kv] = v 1 1 [Fw:Kv] w.

7 ON ARITHMETIC GENERAL THEOREMS FOR POLARIZED VARIETIES 7 Then w,k is an absolute value on F which represents w and extends v. We often identify v and w and write: (2.1) v,k := w,k = N Fw/K v ( ) [Fw:Kv] = v 1 1 [Fw:Kv] v. Finally, in case that K = k(y) is a function field, then F = k(y ) f Y Y the nmalization of Y in F. In this context, M F = M (Y,L ), f L the pullback of L to Y Weil and Proximity functions f Cartier diviss. Let X be a geometrically irreducible projective variety over K. We consider local Weil functions determined by meromphic sections of P, a line bundle on X defined over F. Fix global sections s 0,...,s n and t 0,...,t m of globally generated line bundles M and N on X, defined over F, and so that P M N 1. Fix s a meromphic section of P. As in [3], we say that D = D(s) = (s;m,s;n,t) is a presentation of P with respect to s (defined over F). We define the local Weil function of P, with respect to D and a fixed place v M K, by: 1 (2.2) λ D (y,v) := maxminlog s k k l t l s (y) = maxminlog s k k l v,k t l s (y) [Fw:Kv]. This function has domain X(K) \ Supp(s)(K). In (2.2), the place v is extended to F and the absolute value v,k is defined as in (2.1). Some basic properties of local Weil functions can be deduced along the lines of [3, Section 2.2] and [18, Chapter 10]. At times, we identify a meromphic section s of P with the Cartier divis D = div(s) on X F that it determines. In this context, we also say that D is a Cartier divis on X, defined over F. We also use the notations λ D,v ( ), λ s,v ( ) and λ D(s) (,v) to denote the local Weil function λ D (,v). We define the global Weil function f D, nmalized relative to K, by: λ D ( ) = v M K λ D ( ;v). ThisfunctionhasdomainthesetofK-pointsx X(K)outsideofthesupptofD = div(s). The proximity function of D with respect to a finite set S M K of places of K also has domain X(K)\Supp(D)(K). It is defined by: m S (,D) := λ D,v ( ). v

8 8 NATHAN GRIEVE 2.3. Weil functions f subschemes. Similar to [28], via Weil functions of the exceptional divis E of π: X XF, the blowing-up of X along a proper subscheme Y X, defined over F, we have concepts of λ Y,v ( ), the Weil function of Y with respect to places v S. Me precisely, we identify E with a line bundle on X and defined over F. We then define the Weil function f Y with respect to places v of K (extended to F) via: λ Y,v ( ) := λ E,v ( ); such functions have domain the set of K-points of X outside of the suppt of Y Logarithmic height functions. Let h L ( ) denote the logarithmic height of a line bundle L on X, defined over K. This function is nmalized relative to the base field K. When L is very ample, h L ( ) is obtained by pulling back the standard logarithmic height function on projective space P n K : h OP n(1)(x) = maxlog x j v, j v M K f x = (x 0,...,x n ) P n (K) and n = h 0 (X,L) 1. In general, h L ( ) is obtained by first expressing L as a difference of very ample line bundles L M N. Then h L ( ) is defined, up to equivalence, as the difference of h M ( ) and h N ( ) Schmidt s Subspace Theem. F later use, see Theem 4.1, we require an extension of Schmidt s Subspace Theem. Proposition 2.1 (Compare with [26, Theem 2.7]). Let X be a geometrically irreducible projective variety over K and L an effective line bundle on X, defined over K. Let 0 V H 0 (X,L) be an effective linear system, put n := dim V, and f each place v S, fix a collection of sections: σ 1,v,...,σ q,v V F := F K V H 0 (X F,L F ), with Weil functions λ σi,v,v( ). Let ǫ > 0. Then there exists positive constants a ǫ and b ǫ together with a proper Zariski closed subset Z X so that f all x X(K)\Z(K), either max J h L (x) a ǫ ; λ σj,v,v(x) (ǫ+n+1)h L (x)+b ǫ. j J Here the maximum is taken over all subsets J {1,...,q} f which the sections σ j,v, with j J, are linearly independent.

9 ON ARITHMETIC GENERAL THEOREMS FOR POLARIZED VARIETIES 9 Remark 2.2. In Proposition 2.1, the subvariety Z is essentially determined by the K-span of a finite collection of hyperplane sections. Proof of Proposition 2.1. Fix v S. Each basis σ 0,v,...,σ n,v V F H 0 (X F,L F ) determines a rational map φ VF,v: X F P n F. By elimination of indeterminacy of rational maps, we obtain a commutative diagram of F-schemes: π 1 (U F ) π XF φṽf U F X F P n F. In particular, φ UF X F = X Spec(K) Spec(F), f some K-variety X, independent of v S. The mphism π is projective and there exists effective line bundles M and B on X, defined over K, with the properties that: M F φ Ṽ F O P n F (1) = π L F B F and π σ iv H 0 ( X,M F ), f each i = 0,...,n and each v S. Consider the Weil functions λ π σ i,v,v( ) f M with respect to the global sections π σ i,v, the pullback of the Weil functions λ σi,v,v( ) f L and the relation: (2.3) π L F M F +B F. It follows from (2.3) that the logarithmic heights of the line bundles π L, M and B are related by: (2.4) h π L( ) = h M ( )+h B ( )+O(1). Furtherme, the local Weil functions are related via: (2.5) π λ σi,v( ) = λ π σ i,v,v( )+λ B,v ( )+O(1). Here, λ B,v ( ) = λ B (,v) is a fixed Weil function f B, depending on the choice of a meromphic section. The conclusion desired by Proposition 2.1 can now be deduced by combining the equations (2.4) and (2.5) together with the version of Schmidt s Subspace Theem as described f instance in [3, Theem 7.2.2] [15, Theem 5.2].

10 10 NATHAN GRIEVE 3. Amplitude of Jets and the Filtration construction of Autissier Here we discuss complexity of filtered linear series. The main point is to give a unified interpretation of concepts of growth of filtered series which appear in [1], [2], [23] and [26] LetLbeabiglinebundleonX, anirreducibleprojectivevarietyoverk, analgebraically closed field of characteristic zero, with section ring R(L) = R(X,L) = m 0H 0 (X,mL). The vanishing numbers of a decreasing, multiplicative R-filtration F = F R(L) of R(L) are defined, f each m 0, by the condition that: a j (ml) = inf { t R : codimf t H 0 (X,mL) j +1 }. Such filtrations determine discrete measures on the real line: 1 ν m,f = ν m := h 0 (X,mL) f each m 0. j δ m 1 a j (ml), 3.2. Let E be an effective line bundle on X. Put: l 1 α(l,e) := h0 (X,L le) h 0 (X,L) and: (3.1) β(l,e) = liminf m m 1 α(ml,e) = liminf m F later use, set: [26] [2], and [23]. γ(l,e) = β(l,e) 1, β E (L) := 0 Vol(L te) dt, Vol(L) l 1 h0 (X,mL le). mh 0 (X,mL) 3.3. SupposethatX isnmalandthatlisabiglinebundleonx. Weestablishequivalence of the quantities β(l,e) and β E (L). Proposition 3.1. If X is nmal and if L is a big line bundle on X, then: (3.2) β(l,e) = β E (L).

11 ON ARITHMETIC GENERAL THEOREMS FOR POLARIZED VARIETIES 11 Proof. Let F = F R(L) be the filtration of R(L) induced by ders of vanishing along E, with vanishing numbers a j (ml) and discrete measures ν m. Put: and As in [4, Theem 1.11], we can write: E(ν m ) = 0 h m (t) := h0 (X,mL mte) h 0 (X,mL) E(ν m ) := t d(ν m ) = 0 Further, we know from [4, Theem 1.11] that: 0 t d(ν m ). (3.3) β E (L) = lim m E(ν m). t h m (t)dt = h m (t)dt. The functions h m (t) are also constant on intervals of width m. Thus: 1 (3.4) h m (t)dt = h 0 (X,mL le). mh 0 (X,mL) Rewriting (3.4), we get: (3.5) 0 0 h m (t)dt = m 1 + l=0 m 1 h 0 (X,mL) h 0 (X,mL le). Because of (3.3) and dominated convergence, (3.5) implies, as m : l=1 (3.6) β E (L) = E(ν) = lim h0 (X,mL le). m mh 0 (X,mL) l= When L is very ample, there is also an interpretation as a nmalized Chow weight. Collary 3.2. If X is nmal and L is very ample, then: e X (c) (3.7) β(l,e) = β E (L) = (dimx +1)(deg L X) = lim l 1 h0 (X,mL le). m mh 0 (X,mL) Here e X (c) is the Chow weight of X in P n k with respect to c = (a 0 (L),...,a n (L)), the weight vect of L along E, and the embedding of X into P n k is induced by a basis of H 0 (X,L) which is compatible with the filtration given by ders of vanishing along E.

12 12 NATHAN GRIEVE Proof. We combine (3.1), (3.2) and results from [16]. F instance, a key point, [16, Proof of Proposition 4.1], is: (3.8) E(ν m ) = s(m, c) mh 0 (X,mL). In (3.8), s(m,c) denotes the mth Hilbert weight of X in P n k with respect to the weight vect c = (a 0 (L),...,a n (L)) We now indicate how the filtration construction of Cvaja-Zannier, [7], Levin, [21], Autissier, [2], and Ru-Vojta, [26], relates to Proposition 3.1 and Collary 3.2. Fix effective Cartier diviss D 1,...,D q on X. We assume that these diviss intersect properly and put: { } Σ := σ {1,...,q} : j σsuppd j. Fix σ Σ and, f each integer b 0, let: { σ = σ (b) := a = (a i ) N #σ : i σ a i = b }. Note that the set σ is finite F each t R 0 and each a σ, define the ideal sheaf I(t;a;σ) by: ( I(t;a;σ) = O X ) b i D i. i σ b N #σ i σ a ib i bt Note also that the ideal sheaf I(t;a;σ) is generated by finitely many such b In this way, we obtain a decreasing, multiplicative R-filtration of R(L). Such filtrations have the fm: F t (m;σ;a) = H 0 (X,mL I(t;a;σ)) H 0 (X,mL), f m 0. If then we let 0 s H 0 (X,mL), µ a (s) = µ(s) = sup{t R 0 : s F t (m;σ;a)} = max{t R 0 : s F t (m;σ;a)}. In what follows, we denote by B σ;a,m = B σ;a a basis f H 0 (X,mL) which is adapted to the filtration F (m;σ;a).

13 ON ARITHMETIC GENERAL THEOREMS FOR POLARIZED VARIETIES Fix an integer m 0 and let a min (m;σ;a) = a 0 (m;σ,a)... a nm (m;σ;a) = a max (m;σ;a) denote the vanishing numbers of the filtrations F (m;σ;a) = (F t (m;σ;a)) t R 0. The discrete measures on R that they determine are then: 1 ν(m;σ;a) = h 0 (X,mL) let E(m; σ; a) denote the expectations of such measures: E(m;σ;a) := 0 j δ m 1 a j (m;σ;a); t ν(m;σ;a)dt. Proposition 3.3. Let L be a big line bundle on X. The expectations E(m;σ;a) have the properties that: ( ) 1 E(m;σ;a) mh 0 (X,mL) min h 0 (X,mL ld i ). 1 i q l 1 Proof. This is a refmulation of [26, Proposition 6.7]. In me detail, since the basis B σ;a;m is adapted to the filtration, it follows that: 1 (3.9) E(m;σ;a) = µ mh 0 a (s). (X,mL) s B σ;a;m On the other hand, by [26, Proposition 6.7], we have: ( 1 (3.10) h 0 (X,mL) µ a (s) min i s B σ;a;m Thus, combining, (3.9) and (3.10), we obtain: ( E(m;σ;a) min i 1 mh 0 (X,mL) 1 h 0 (X,mL) ) h 0 (X,mL ld i ). l 1 ) h 0 (X,mL ld i ). l We mention that the righthand side of Proposition 3.3 can be interpreted in terms of the Hilbert weights s(m,c i ) of X in P n k, n = h0 (X,L) 1, with respect to the weight vects c i determined by the filtrations of H 0 (X,mL) induced by the diviss D i. One description of the Hilbert weights, is given in [16, Section 4].

14 14 NATHAN GRIEVE Collary 3.4. Let L be a very ample line bundle on X. Then: ( ) s(m,ci ) E(m;σ;a) min. i mh 0 (X,mL) Proof. We simply observe: α(ml,d i ) = s(m,c i) h 0 (X,mL) Fix a nonzero section 0 s H 0 (X,mL) and consider the finite set K = K σ,a,s that consists of minimal elements of the set: { b N #σ : i σ a i b i bµ a (s) } In this notation, we then have that: L m I(µ a (s)) = ( ml b K i σ b i D i ) One other imptant point to observe is that, as is a consequence of [26, Proposition 4.18], (3.11) div(s) ( ) b i D i. b K i σ In (3.11), it is meant that we identify the Cartier divis div(s) with its pullback to a nmal proper model of X that realizes the right hand side as a Cartier divis Furtherme, [26, Lemma 6.8], which uses (3.11), establishes that: (3.12) ( ) div(s) b min h 0 (X,mL ld i ) D, b+d 1 i q s B σ;a σ Σ a σ f d = dimx, the dimension of X. Similarly, in (3.12), we also identify the Cartier divis D withits pullback to a nmal proper model of X that realizes the lefthandside asacartier divis. l 1

15 ON ARITHMETIC GENERAL THEOREMS FOR POLARIZED VARIETIES The Arithmetic General Theem In this section, we study and extend the Arithmetic General Theem of Ru and Vojta, [26]. In doing so, we prove Theem 4.1. In Section 5, we note that it implies Theem In proving Theem 4.1 below, our argument is based on [26, proof of Arithmetic General Theem]. Theem 4.1 (Compare with [26, Arithmetic General Theem]). Let X be a geometrically irreducible projective variety over K, a number field characteristic zero function field, and let D 1,...,D q be nonzero effective Cartier diviss on X, defined over F, which intersect properly. Let L be a big line bundle on X, defined over K. Then, f each ǫ > 0, there exists constants a ǫ, b ǫ > 0 with the property that either: m S (x,d) h L (x) a ǫ ; ( ) max γ(l,d j)+ǫ h L (x)+b ǫ 1 j q f all K-rational points x X(K) outside of some proper Zariski closed subset Z X In our proof of Theem 4.1, to reduce notation, we denote, with extensive abuse of notation, by X the base change of X with respect to the field extension F/K. We also denote by L the pullback of L via this base change. When no confusion is likely, we make no explicit mention about allowing coefficients in F, as opposed to only allowing coefficients in K. We also emphasize that our proof relies on Proposition 3.3. F example, it requires that the diviss D 1,...,D q, which are defined over F, intersect properly (over F) We now proceed to prove our Main Arithmetic General Theem. Proof of Theem 4.1. Let d denote the dimension of X. Let ǫ > 0 and choose a real number ǫ 2 > 0 together with positive integers m and b so that: ( (4.1) 1+ d ) ( ) mh 0 (X,mL)+mǫ max 2 < max b 1 i q l 1 h0 (X,mL ld i ) γ(l,d i)+ǫ. 1 i q Write: F each i = 1,...,T 1, let be the subset such that B σ;a = B 1 BT2 = {s 1,...,s T2 }. σ;a J i {1,...,T 2 } B i = {s j : j J i }.

16 16 NATHAN GRIEVE Let B i denote the divis ( ) B i := div div(s j ) ; j B i choose Weil functions λ D,v ( ), λ Bi,v( ) and λ sj,v( ), f each v S, and each j = 1,...,T 2. The key point then is that, because of (3.12) and [26, Proposition 4.10] (parts (b) and (c) can be adapted using the general they of [18, Chapter 10] so as to apply to our current setting), f each v S, it also holds true that: But: and so: max λ Bi,v( )+O v (1) b 1 i T 1 b+d ( b min d+b 1 i q ( min 1 i q ) h 0 (X,mL ld i ) λ D,v ( ). l 1 max λ Bi,v( )+O v (1) = max λ sj,v( )+O v (1) 1 i T 1 1 i T 1 j J i ) h 0 (X,mL ld i ) l 1 λ D,v ( ) max λ sj,v( )+O v (1). 1 i T 1 j J i We now apply Schmidt s Subspace Theem (in the fm of Proposition 2.1). In particular, we may apply that remark to the sections s j since they are sections of L (a prii with coefficients in F) and since the big line bundle L is defined over the base field K. OurconclusionthenisthatthereexistsaproperZariskiclosedsubsetZ X andconstants a ǫ2, b ǫ2 so that f all K-points x X \Z either: max J h ml (x) a ǫ2 λ sj,v(x) ( ) h 0 (X,mL)+ǫ 2 hml (x)+b ǫ2. j J Here, the maximum is taken over all subsets J {1,...,T 2 } f which the sections s j, f j J, are linearly independent. In particular, either: λ D,v (x) ( 1+ d ) max b 1 i q f all K-points x X \Z. Finally, since h ml (x) a ǫ2 ( ) h 0 (X,mL)+ǫ 2 h ml (x)+b l 1 h0 ǫ (X,mL ld i ) 2 h ml (x) = mh L (x),

17 ON ARITHMETIC GENERAL THEOREMS FOR POLARIZED VARIETIES 17 we obtain that either: λ D,v (x) h L (x) A ǫ ( ) max γ(l,d i)+ǫ h L (x)+b ǫ 1 i q f positive constants A ǫ, B ǫ (depending on ǫ) and all K-points x X \Z. 5. Proof of Theem 1.1, Collary 1.2 and Collary 1.3 In this section, we prove Theem 1.1, Collary 1.2 and Collary First we prove Theem 1.1. Proof of Theem 1.1. Combine Theem 4.1 and Proposition Next, we prove Collary 1.2. Proof of Collary 1.2. Collary 1.2 is a consequence of Theem 1.1 applied to X Finally, we note that Collary 1.3 is a special case of Theem 5.1 below. Theem 5.1. Suppose that X is a geometrically irreducible Q-Genstein geometrically nmal variety, defined over K, and assume that the Q-Cartier divis K X is Q-ample. Let Y X be a proper subscheme, defined over F, and write the exceptional divis E of the blowing-up of X along Y, in the fm π: X XF, E = E E q, f E i effective Cartier diviss on X. In this context, assume that the diviss E 1,...,E q intersect properly and β( π K X,E i ) 1 f all i. Then Vojta s inequalities hold true. Precisely, if M is a big line bundle on X, defined over K, then f all ǫ > 0, there exists constants a ǫ,b ǫ > 0 so that either: h KX (x) a ǫ λ Y,v (x)+h KX (x) ǫh M (x)+b ǫ f all K-rational points x X(K) outside of some proper subvariety of X.

18 18 NATHAN GRIEVE Proof of Theem 5.1 and Collary 1.3. Withoutlossofgenerality, wemayassume thatthe ample Cartier divis K X is integral. We first establish the well-known reduction step. If Vojta s inequalities hold true f M = K X and all ǫ > 0, then the same is true f all big line bundles M (perhaps after adjusting ǫ and the proper subvariety). To this end, as in [19, Collary 2.2.7], since K X is ample and M is big, there exists m > 0 and an effective line bundle N on X so that: Thus: and: mm lin K X +N. mh M ( ) = h KX ( )+h N ( )+O(1); (5.1) h KX ( ) mh M ( )+O(1) outside of Bs(N), the base locus of N. Let ǫ > 0. We show that: (5.2) λ Y,v (x)+h KX (x) ǫh M (x)+o(1) f all x X(K) outside of some proper Zariski closed subset Z X. Note: (5.3) ǫh KX ( ) ǫmh M ( )+O(1) outside of Bs(N); put: We can rewrite (5.3) as: ǫ = ǫ m. (5.4) ǫ h KX ( ) ǫh M ( )+O(1) outside of Bs(N). On the other hand, by assumption: (5.5) λ Y,v (x)+h KX (x) ǫ h KX (x)+o(1) f all x X(K) outside of some Zariski closed subset W X depending on ǫ. Combining (5.4) and (5.5), we then have: (5.6) λ Y,v ( )+h KX (x) ǫ h KX (x)+o(1) ǫh M (x)+o(1), f all x X(K) outside of The inequality (5.2) is thus implied by (5.6). Z := W Bs(N). Now put M = K X and let ǫ > 0. We can rewrite the inequality λ Y,v ( )+h KX ( ) ǫh KX ( )+O(1)

19 in terms of the blowing-up of X along Y: ON ARITHMETIC GENERAL THEOREMS FOR POLARIZED VARIETIES 19 π: X XF (5.7) m S (,E) (1+ǫ)h π K X ( )+O(1). By assumption, E = E E q, the Cartier diviss E i intersect properly and f all i. Thus: β( π K X,E i ) 1, max 1 i q β( π K X,E i ) 1 +ǫ 1+ǫ. We now apply Theem 4.1 and Proposition 3.1. There exists constants ã ǫ, b ǫ > 0 so that either: (5.8) h π K X (x) ã ǫ ; (5.9) m S (x,e) ( ) max 1 i q β( π K X,E i ) 1 +ǫ h π K X (x)+ b ǫ f all K-rational points x X(K) outside of some proper Zariski closed subset Z X. We can rewrite these inequalities (5.8) and (5.9) in terms of X. As in (5.6) and (5.7), we obtain constants a ǫ,b ǫ > 0 so that either: h KX (x) a ǫ λ Y,v (x) (1+ǫ)h KX (x)+b ǫ f all K-rational points x X(K) outside of some proper subvariety Z X. References [1] P. Autissier, Géométries, points entiers et courbes entières, Ann. Sci. Éc. Nm. Supér. (4) 42 (2009), no. 2, [2] P. Autissier, Sur la non-densité des points entiers, Duke Math. J. 158 (2011), no. 1, [3] E. Bombieri and W. Gubler, Heights in Diophantine geometry, Cambridge University Press, Cambridge, [4] S. Boucksom and H. Chen, Okounkov bodies of filtered linear series, Compos. Math. 147 (2011), no. 4, [5] S. Boucksom, T. Hisamoto, and M. Jonsson, Unifm K-stability, Duistermaat-Heckman measures and singularities of pairs, Ann. Inst. Fourier (Grenoble) 67 (2017), no. 2, [6] S. Boucksom, A. Küronya, C. Maclean, and T. Szemberg, Vanishing sequences and Okounkov bodies, Math. Ann. 361 (2015),

20 20 NATHAN GRIEVE [7] P. Cvaja and U. Zannier, A subspace theem approach to integral points on curves, C. R. Math. Acad. Sci. Paris 334 (2002), no. 4, [8] P. Cvaja and U. Zannier, On a general Thue s equation, Amer. J. Math. 126 (2004), no. 5, [9] P. Cvajaand U. Zannier, On integral points on surfaces, Ann. of Math. (2) 160 (2004), no. 2, [10] S. K. Donaldson, Scalar curvature and stability of tic varieties, J. Differential Geom. 62 (2002), no. 2, [11] J.-H. Evertse and R. G. Ferretti, Diophantine inequalities on projective varieties, Int. Math. Res. Not. (2002), no. 25, [12] G. Faltings and G. Wüstholz, Diophantine approximations on projective spaces, Invent. Math. 116 (1994), [13] R. G. Ferretti, Mumfd s degree of contact and diophantine approximations, Compos. Math. 121(2000), [14] R. G. Ferretti, Diophantine approximations and tic defmations, Duke Math. J. 118 (2003), no. 3, [15] N. Grieve, Diophantine approximation constants f varieties over function fields, Michigan Math. J. 67 (2018), no. 2, [16] N. Grieve, Expectations, concave transfms, Chow weights, and Roth s theem f varieties, Preprint (2018). [17] G. Heier and A. Levin, A generalized Schmidt subspace theem f closed subschemes, arxiv: v1. [18] S. Lang, Fundamentals of Diophantine geometry, Springer-Verlag, New Yk, [19] R. Lazarsfeld, Positivity in algebraic geometry I, Springer-Verlag, Berlin, [20] R. Lazarsfeld and M. Mustaţǎ, Convex bodies associated to linear series, Ann. Sci. Éc. Nm. Supér. (4) 42 (2009), no. 5, [21] A. Levin, Generalizations of Siegel s and Picard s theems, Ann. of Math. (2) 170 (2009), no. 2, [22] A. Levin, On the Schmidt subspace theem f algebraic points, Duke Math. J. 163 (2014), no. 15, [23] D. McKinnon and M. Roth, Seshadri constants, diophantine approximation, and Roth s theem f arbitrary varieties, Invent. Math. 200 (2015), no. 2, [24] D. McKinnon and M. Roth, An analogue of Liouville s theem and an application to cubic surfaces, Eur. J. Math 2 (2016), no. 4, [25] M. Ru, On a general Diophantine inequality, Funct. Approx. Comment. Math. 56(2017), no. 2, [26] M. Ru and P. Vojta, A birational Nevanlinna constant and its consequences, Final Version (2018). [27] M. Ru and J. T.-Y. Wang, A Subspace Theem f Subvarieties, Algebra Number They 11 (2017), no. 10, [28] J. H. Silverman, Arithmetic Distance Functions and Height Functions in Diophantine Geometry, Math. Ann. 279 (1987), [29] P. Vojta, Diophantine approximations and value distribution they, Springer-Verlag, Berlin, Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA address: grievena@msu.edu