European Journal of Mathematics

, Volume 2, Issue 4, pp 929–959 | Cite as

An analogue of Liouville’s Theorem and an application to cubic surfaces

Research Article

Abstract

We prove a strong analogue of Liouville’s Theorem in Diophantine approximation for points on arbitrary algebraic varieties. We use this theorem to prove a conjecture of the first author for cubic surfaces in \(\mathbb {P}^3\).

Keywords

Seshadri constants Diophantine approximation Rational points 

Mathematics Subject Classification

14G05 14G40 

1 Introduction

The famous theorem of K.F. Roth (see for example [5, Part D]) gives a sharp upper bound on how well an irrational algebraic number can be approximated by rational numbers. In [10], the authors prove an analogue of Roth’s Theorem for algebraic points on arbitrary algebraic varieties. In this paper we generalize, in the sense of [10], Liouville’s Approximation Theorem to arbitrary varieties, as well as giving an extension involving the stable base locus.

The point of view of [10] is that Roth’s and Liouville’s Theorems are examples of “local Bombieri–Lang phenomena” whereby local positivity of a line bundle influences local accumulation of rational points. Specifically, given a variety X, an algebraic point \(x\in X\), and an ample line bundle L on X, these theorems are expressed as inequalities between \(\epsilon _{x}(L)\), the Seshadri constant, measuring local positivity of L near x, and \(\alpha _{x}(L)\), an invariant measuring how well we can approximate x by rational points.

Roth’s Theorem is usually thought of as stronger than Liouville’s, but if the locus being approximated is defined over the ground field, Liouville’s Theorem is strictly better. On \(\mathbb {P}^1\) one gains a factor of two. For arbitrary varieties, however, moving past the Seshadri constant into the non-nef part of the big cone can provide even larger gains. We need this improvement for our application in Sect. 4 where we verify a conjecture of the first author for cubic surfaces in \(\mathbb {P}^3\).

In Sect. 2 we review the definitions and elementary properties of \(\alpha _{x}\) and \(\epsilon _{x}\). In Sect. 3 we prove the generalized Liouville Theorem (Theorem 3.3). We close the paper in Sect. 4 by computing \(\alpha _{x}\) and \(\epsilon _{x}\) for an arbitrary nef line bundle and rational point, not on a line, on a smooth cubic surface (where the lines are also rational); we then use this to verify [9, Conjecture 3.2].

2 Elementary properties of \(\alpha \) and \(\epsilon \)

In this section, we give a brief overview of the properties of \(\alpha \) and \(\epsilon \) used in this paper. For a more detailed discussion of \(\alpha \), see [10]. For a more detailed discussion of \(\epsilon \), there are many good references—see for example [7, Chapter 5]. Proofs of all of the facts listed below can be found in [10].

Let k be a number field, and X a projective variety over \(\mathrm{Spec}(k)\) (i.e., a reduced subscheme of some finite-dimensional projective space Open image in new window).

The constant\(\alpha _{x}\). In order to motivate the definition of \(\alpha _{x}\) it is helpful to recall the classical case of approximation on the line. For a point \(x\in \mathbb {R}\) the approximation exponent\(\tau _{x}\) of x is the unique extended real number \(\tau _{x}\in (0,\infty ]\) such that the inequalityhas only finitely many solutions \(a/b\in \mathbb {Q}\) whenever \(\delta >0\) (respectively has infinitely solutions \(a/b\in \mathbb {Q}\) whenever \(\delta <0\)). The approximation exponent measures a certain tension between our ability to closely approximate x by rational numbers (the distance term Open image in new window) and the complexity (the 1 / b term) of the number required to make this approximation. In this notation the 1844 theorem of Liouville [8] is that \(\tau _{x}\leqslant d\) for \(x\in \mathbb {R}\) algebraic of degree d over \(\mathbb {Q}\).
To generalize \(\tau _x\) to arbitrary projective varieties over \(\mathrm{Spec}(k)\) for a number field k, we replace the function Open image in new window by a distance function Open image in new window depending on a place v of k, and measure the complexity of a point via a height function Open image in new window depending on an ample line bundle L. For an introduction to the theory of heights the reader is referred to any one of [2, Chapter 2], [5, Part B], [6, Chapter III], or [12, Chapter 2]. Unless otherwise specified all height functions in this paper are multiplicative, relative to k, and come from line bundles on X defined over k. In this paper we use the following normalizations. The absolute values are normalized with respect to k: if v is a finite place of k, \(\pi \) a uniformizer of the corresponding maximal ideal, and \(\kappa \) the residue field then \(\Vert \pi \Vert _{v}=1/\#\,\kappa \); if v is an infinite place corresponding to an embedding \(i:k\hookrightarrow \mathbb {C}\) then Open image in new window for all \(x\in k\), where \(m_v=1\) or 2 depending on whether v is real or complex. The heights are then normalized so that for a point Open image in new window, the height with respect to \(\mathscr {O}_{\mathbb {P}^n}(1)\) iswhere the product ranges over all the places v of k.

In order to define a distance function we fix a place v of k and extension (which we also call v) to \(\overline{k}\).

If vis archimedean.  We choose a distance function on \(X(\overline{k})\) by choosing an embedding Open image in new window and pulling back (via v) the function on Open image in new window given by the formulawhere Open image in new window and Open image in new window are points of Open image in new window, and Open image in new window is the absolute value on \(\mathbb {C}\) extending the usual absolute value on \(\mathbb {R}\), i.e., such that Open image in new window.
If vis non-archimedean.  We choose a distance function on \(X(\overline{k})\) by choosing an embedding Open image in new window and pulling back the distance function on Open image in new window given by the formulawhere Open image in new window and Open image in new window are points of Open image in new window.

These definitions are standard in Arakelov theory, albeit here we have normalized with respect to k rather than \(\mathbb {Q}\). (See for instance [2, Section 2.8] where a distance function Open image in new window is defined for each place v; the distance functions are related by Open image in new window.)

Two real valued functions g and \(g'\) with the same domain are called equivalent if there are positive constants \(c\leqslant C\) such that Open image in new window for all values of the domain. The distance functions defined above depend on the choice of embedding into projective space, but by [10, Proposition 2.4] any two embeddings give equivalent distance functions on Open image in new window. (We may restrict ourselves to points of \(X(k_v)\) since any point of \(X(\overline{k})\) that does not lie in \(X(k_v)\) cannot be approximated by k-rational points; see also Remark 2.7.) It follows from the definition of \(\alpha _{x}\) below that equivalent distance functions produce the same value of \(\alpha _{x}\); thus our definition of \(\alpha _{x}\) does not depend on the projective embedding chosen to define \(d_v\). A more geometric definition of distance, and a proof that it is equivalant to the distance formulae above, may be found in Sect. 6.

Definition 2.1

Let X be a projective variety, \(x\in X(\overline{k})\), L a line bundle on X. For any sequence \(\{x_i\}\subset X(k)\) of distinct points with \(d_v(x,x_i)\rightarrow 0\) (which we denote by \(\{x_i\}\rightarrow x\)), we set

Remark 2.2

(a) It follows easily from the definition that if \(A(\{x_i\}, L)\) is nonempty then it is an interval unbounded to the right, i.e., if \(\gamma \in A(\{x_i\},L)\) then \(\gamma +\delta \in A(\{x_i\},L)\) for any \(\delta >0\).

(b) If \(\{x_i'\}\) is a subsequence of \(\{x_i\}\) then \(A(\{x_i\},L)\subseteq A(\{x_i'\},L)\).

Definition 2.3

If \(A(\{x_i\},L)\) is empty we set \(\alpha _{x}(\{x_i\},L)=\infty \). Otherwise we set \(\alpha _{x}(\{x_i\}, L)\) to be the infimum of \(A(\{x_i\},L)\). We call \(\alpha _{x}(\{x_i\},L)\) the approximation constant of \(\{x_i\}\) with respect to L.

As \(i\rightarrow \infty \) we have \(d_v(x,x_i)\rightarrow 0\). We thus expect that \(d_v(x,x_i)^{\gamma }H_{L}(x_i)\) goes to 0 for large \(\gamma \) and to \(\infty \) for small \(\gamma \). The number \(\alpha _{x}(\{x_i\},L)\) marks the transition point between these two behaviours.

By Remark 2.2 (b) if \(\{x_i'\}\) is a subsequence of \(\{x_i\}\) then \(\alpha _{x}(\{x_i'\},L)\leqslant \alpha _{x}(\{x_i\},L)\). Thus we may freely replace a sequence with a subsequence when trying to establish lower bounds.

Definition 2.4

Let k be a number field, X a projective variety over \(\mathrm{Spec}(k)\), L a line bundle on X, and \(x\in X(\overline{k})\). Then \(\alpha _{x}(L)\) is defined to be the infimum of all approximation constants of sequences of points in X(k) converging to x. If no such sequence exists then set \(\alpha _{x}(L)=\infty \).

To see the connection with the usual approximation exponent on \(\mathbb {P}^1\), suppose that L is an ample line bundle. We may define an approximation constant \(\tau _x(L)\) by simply extending the definition on \(\mathbb {P}^1\), namely by defining \(\tau _{x}(L)\) to be the unique extended real number \(\tau _x(L)\in [0,\infty ]\) such that the inequality
$$\begin{aligned} d_v(x,y) < \frac{1}{H_{L}(y)^{\tau _x(L)+\delta }} \end{aligned}$$
has only finitely many solutions \(y\in X(k)\) whenever \(\delta >0\) (respectively has infinitely many solutions \(y\in X(k)\) whenever \(\delta <0\)). Then [10, Proposition 2.11] implies that \(\alpha _{x}(L) = 1/\tau _x(L)\). In particular the theorem of Liouville becomes \(\alpha _{x}(\mathscr {O}_{\mathbb {P}^1}(1))\geqslant 1/d\) for \(x\in \mathbb {R}\) of degree d over \(\mathbb {Q}\), and it is this type of lower bound that we wish to generalize to arbitrary varieties. The choice of using the reciprocal of \(\tau \) is justified by the resulting formal similarity with the Seshadri constant, and more natural behaviour when we vary L (see, for example, Proposition 2.11).

We need two results on \(\alpha _{x}\) before continuing onto the Seshadri constant. First, we will need to know how to calculate \(\alpha _{x}\) in one simple case.

Lemma 2.5

([10, Lemma 2.13]) Let Open image in new window. Then \(\alpha _{x,\mathbb {P}^n}({\mathscr {O}}_{\mathbb {P}^n}(1))=1\).

Second, it will be useful to know how the approximation constant changes when we change the field k. Let K / k be a finite extension with \(K\subset \overline{k}\), and set Open image in new window. A point \(y\in X(\overline{k})\) is a map \(\mathrm{Spec}(\overline{k})\rightarrow X\) over \(\mathrm{Spec}(k)\), and factors through the map \(X_{K}\rightarrow X\), i.e., such a point x gives a point of \(X_{K}(\overline{k})\). We thus have a canonical identification \(X_{K}(\overline{k})=X(\overline{k})\) which we use implicitly in the paragraphs below. We use the notation that \(\alpha _{x}(\{x_i\},L)_{K}\) (respectively \(\alpha _{x}(L)_{K}\)) denotes the approximation constant of a sequence (respectively point x) computed on \(X_{K}\) with respect to K. This means that when computing \(\alpha \), we use the height \(H_{L}\) relative to K and normalize \(d_v\) relative to K (we define \(d_v\) on \(X_K\) using the same embedding used to define \(d_v\) on X). If Open image in new window and Open image in new window (where \(K_v\) and \(k_v\) denote the completions of K and k with respect to v) then this means simply that \(H_{L}(x_i)_{K} = H_{L}(x_i)^{d}_{k}\) and \(d_v(x,x_i)_{K} = d_v(x,x_i)^{m_v}_{k}\).

Proposition 2.6

Let X be a variety over \(\mathrm{Spec}(k)\), \(x\in X(\overline{k})\) any point, L a line bundle on X, and \(\{x_i\}\rightarrow x\) a sequence of points in X(k) approximating x. Let K be any finite extension of k and set Open image in new window. The base change of \(\{x_i\}\) gives a sequence \(\{y_i\}\) of K-points of \(X_{K}\) approximating x. Set Open image in new window, and let Open image in new window. Then
$$\begin{aligned} \alpha _x({\{y_i\}, L})_{K} = \frac{d}{m_v} \,\alpha _x({\{x_i\}, L})_{k}. \end{aligned}$$
In particular, we have the bound \(\alpha _x(L)_{K} \leqslant d\alpha _x(L)_{k}/m_v\).

Proof

The claim that \(\alpha _x({\{x_i\}, L})_{K} = d \alpha _x({\{x_i\}, L})_{k}/m_v\) follows immediately from the equalities Open image in new window and Open image in new window. The inequality \(\alpha _x(L)_{K} \leqslant d \alpha _x(L)_{k}/m_v\) then follows since the sequences in \(X_K(K)\) approximating x which come from sequences \(\{x_i\}\) in X(k) are a subset of all the sequences in \(X_K(K)\) approximating x. \(\square \)

Remark 2.7

Let x be a point of \(X(\overline{k})\) and let K be the field of definition of x. If \(K\not \subseteq k_v\), or equivalently, \(K_v\ne k_v\) then it will be impossible to find a sequence of points of X(k) converging (in terms of \(d_v\)) to x. For example, when v is archimedean this happens when \(k_v=\mathbb {R}\) and \(K_v=\mathbb {C}\). Thus, if we can approximate x by points of X(k) we may assume that \(K_v=k_v\) and so \(m_v=1\).

The following result (appearing in [10] as Theorem 2.16 and in [9] as Theorem 2.8, although the latter version is incorrectly stated) is obtained by combining Roth’s and Dirichlet’s Theorems for approximation on \(\mathbb {P}^1\), as well as the local information about the singularity type. It shows how to calculate \(\alpha _{x}\) on any singular k-rational curve.

Theorem 2.8

Let C be any singular k-rational curve and Open image in new window the normalization map. Then for any ample line bundle L on C, and any \(x\in C(\overline{k})\) we have the equalitywhere Open image in new window is the multiplicity of the branch of C through x corresponding to q, and
$$\begin{aligned} r_{q}= {\left\{ \begin{array}{ll} \,0 &{} \quad \text {if}\quad \kappa (q)\not \subseteq k_v,\\ \,1 &{}\quad \text {if}\quad \kappa (q)=k,\\ \,2 &{}\quad \text {otherwise.} \end{array}\right. } \end{aligned}$$

Here \(\kappa (q)\) means the residue field of the point q, and we use \(r_q=0\) as a shorthand for \(d/r_{q}m_{q}=\infty \).

The Seshadri constant.  The Seshadri constant was introduced by Demailly in [4] for the purposes of measuring the local positivity of a line bundle.

Definition 2.9

Let X be a projective variety, x a point of X, and L a nef line bundle on X. The Seshadri constant, \(\epsilon _{x}(L)\), is defined to bewhere \(\pi :\widetilde{X}\rightarrow X\) is the blowup of X at x, with exceptional divisor E.

A basic property of the Seshadri constant is that if L is ample (as opposed to just being nef) then Open image in new window is itself ample on \(\widetilde{X}\) for all rational \(\gamma \in (0,\epsilon _x(L))\). An argument for this appears in the original paper of Demailly [4] defining the Seshadri constant1.

In the discussion of Conjecture 4.2 below we will need the following alternate characterization of the Seshadri constant:

Proposition 2.10

([7, Proposition 5.1.5]) With the same setup as in Definition 2.9,where the infimum is taken over all reduced irreducible curves C passing through x.

In order to indicate the parallels between \(\alpha _{x}\) and \(\epsilon _{x}\), and for use below, we list a few of their formal properties here.

Proposition 2.11

Let X be a projective variety over \(\mathrm{Spec}(k)\), \(x\in X(\overline{k})\), and let L be any nef line bundle on X.
  1. (a)

    For any positive integer m, \(\alpha _{x}(mL) = m\alpha _{x}(L)\) and \(\epsilon _{x}(mL)=m\epsilon _{x}(L)\). (Thus \(\alpha \) and \(\epsilon \) also make sense for nef \(\mathbb {Q}\)-divisors.)

     
  2. (b)
    \(\alpha _{x}\) and \(\epsilon _{x}\) are concave functions of L: for any positive rational numbers a and b, and any \(\mathbb {Q}\)-divisors \(L_1\) and \(L_2\) (again defined over k, but with the exception of the case that Open image in new window we have and where for the last inequality we assume that \(L_1\) and \(L_2\) are nef.
     
  3. (c)

    If Z is a subvariety of X over \(\mathrm{Spec}(k)\) then for any point \(z\in Z(\overline{k})\) we have \(\alpha _{z}(L|_Z)\geqslant \alpha _z(L)\) and \(\epsilon _{z}(L|_{Z})\geqslant \epsilon _{z}(L)\).

     
  4. (d)
    If Y is also a variety over \(\mathrm{Spec}(k)\), \(x\in X(k)\), \(y\in Y(k)\) and \(L_X\) and \(L_Y\) are nef line bundles on X and Y respectively then and
     

Note that by \(L_{X}{\boxplus }L_Y\) we mean the line bundle \(\mathrm{pr}_{X}^{*}L_1 + \mathrm{pr}_{Y}^{*}L_2\) on Open image in new window, where \(\mathrm{pr}_{X}\) and \(\mathrm{pr}_{Y}\) are the projections. We prefer additive notation for line bundles since this is in line with the behaviour of \(\alpha _{x}\) and \(\epsilon _{x}\), and hence use \(L_{X}{\boxplus }L_Y\) rather than Open image in new window.

Proof

All the proofs follow from elementary arguments using the definitions. For the statements about \(\alpha _{x}\) see [10, Proposition 2.14], and for the statements about \(\epsilon _{x}\) see [10, Proposition 3.4]. \(\square \)

3 A Liouville lower bound for \(\alpha \)

In this section, as in the previous one, we fix a number field k and let X be a projective variety over \(\mathrm{Spec}(k)\).

Lemma 3.1

Let x be a point of X(k), and \(\pi :\widetilde{X}\rightarrow X\) the blowup of X at x with exceptional divisor E. Choose an embedding Open image in new window so that Open image in new window. Let \(Z_0,\ldots , Z_n\) be the coordinates on Open image in new window and define functions \(u_i\), \(i=1,\ldots , n\), on the open subset where \(Z_0\ne 0\) by \(u_i=Z_i/Z_0\). For each place v of k, define a function \(e_v:X(k_v)\rightarrow \mathbb {R}_{\geqslant 0}\) byThen
  1. (a)

    \(0\leqslant e_v\leqslant 1\) for all places v.

     
  2. (b)

    Open image in new window is equivalent to Open image in new window as a function on \(X(k_v)\).

     
  3. (c)

    For \(y\in X(k)\), \(y\ne x\), we have \(H_{E}(y) = \bigl (\prod _{w} e_{w}(y)\bigr )^{-1}\).

     

Proof

Part (a) is clear from the definition. We now prove (b). By [10, Proposition 2.4], distance functions associated to different projective embeddings of X are equivalent on Open image in new window, so we may assume that \(d_v\) has been defined using the embedding \(\varphi \). If v is non-archimedean, then the formula for the distance function in the non-archimedean case and the fact that x is sent to Open image in new window giveFor \(y\in X(k_v)\), this is equal to \(e_v(y)\).
In the case that v is archimedean, we may further assume that \(k_v=\mathbb {C}\), since the functions to be compared transform the same way under field extensions. From the formula for the distance in the archimedean case and the fact that x is sent to Open image in new window we obtainFor \(y\in U(k_v)\), \(y_0\ne 0\), and Open image in new window for \(j=1,\ldots , r\). Thus \(d_v(x,y) = (\Vert u_1(y)\Vert _v+\cdots +\Vert u_r(y)\Vert _v)/ (1+\Vert u_1(y)\Vert _v+\cdots +\Vert u_r(y)\Vert _v)\); it is then elementary to check thatfor all \(y\in U(k_v)\). For Open image in new window we have \(1=d_v(x,y)= e_v(y)\), and thus Open image in new window is equivalent to Open image in new window on \(X(k_v)\).
In (c) we are considering points \(y\in X(k)\), \(y\ne x\) also to be points of \(\widetilde{X}(k)\) via the birational map \(\pi \). To prove (c) it suffices, by using the functoriality of heights under pullback, to consider the case that Open image in new window. Then the blowup \(\widetilde{\mathbb {P}}^{\,n}\) of Open image in new window at x is a subvariety of Open image in new window and \({\mathscr {O}}_{\widetilde{\mathbb {P}}^n}(E)\) is the restriction of \({\mathscr {O}}_{\mathbb {P}^n\times \mathbb {P}^{n-1}}(1,-1)\) to \(\widetilde{\mathbb {P}}^{\,n}\). From this description of \({\mathscr {O}}_{\widetilde{\mathbb {P}}^n}(E)\) we obtain the formulafrom which (c) follows easily.\(\square \)

For the following lemma, we will need an additional definition. The stable base locus of a line bundle L is the intersection of the base loci of mL as \(m\rightarrow \infty \). For details, see [7, Definition 2.1.20].

Lemma 3.2

Suppose that \(x\in X(k)\) and let \(\pi :\widetilde{X}\rightarrow X\) be the blowup at x with exceptional divisor E. Let L be a line bundle on X and \(\gamma >0\) a rational number such that Open image in new window is in the effective cone of \(\widetilde{X}\). Let \(B'\) be the stable base locus of \(L_{\gamma }\) and set \(B=\pi (B')\).

Then there is a positive real constant M (depending only on x and L) such that for any sequence of k-points \(\{x_i\}\rightarrow x\) with all points of \(\{x_i\}\) outside of B, we haveIn particular, \(\alpha (\{x_i\},L)\geqslant \gamma \).

Proof

Let Open image in new window. Since \(B'\) is the stable base locus of \(L_{\gamma }\) there is a constant c (depending only on x and L) so that \(H_{L_{\gamma }}(y)\geqslant c\) for all \(y\in U(k)\). Applying Lemma 3.1, we then haveBy Lemma 3.1 (b), \(d_v(x,x_i)\) and \(e_{v}(x_i)\) are equivalent functions on X(k) and therefore Open image in new window for some positive constant M, again depending only on x and L.

For any \(\delta >0\) we thus have Open image in new window and so conclude that \(\gamma -\delta \not \in A(\{x_i\},L)\) since Open image in new window as \(i\rightarrow \infty \). Therefore \(\gamma \leqslant \alpha (\{x_i\},L)\). \(\square \)

The main result of this section is the following implication of Lemma 3.2.

Theorem 3.3

(Liouville-type Theorem) Let X be an algebraic variety over \(\mathrm{Spec}(k)\), \(x\in X(\overline{k})\) any point, and set Open image in new window where K is the field of definition of x. Set Open image in new window, let \(\widetilde{X}\) be the blowup of \(X_K\) at x with exceptional divisor E, and set \(\pi \) to be the composite \(\pi :\widetilde{X}\rightarrow X_K\rightarrow X\). Let L be a nef line bundle on X, and \(\gamma >0\) a rational number such that Open image in new window is in the effective cone of \(\widetilde{X}\). Finally let \(B'\) be the stable base locus of \(L_{\gamma }\) and set \(B=\pi (B')\). Then there is a positive real constant M such that for all \(y\in X(k){\setminus }B(k)\), we have Open image in new window, and
  1. (a)

    For any sequence \(\{x_i\}\rightarrow x\) of k-points approximating x, if infinitely many points of \(\{x_i\}\) are outside B then \(\alpha (\{x_i\},L) \geqslant \gamma /d\).

     
  2. (b)

    If \(\alpha _{x}(L) < \gamma /d\) then \(x\in B\) and \(\alpha _{x}(L) = \alpha _{x}(L|_B)\).

     
  3. (c)

    If \(x\in B\) and \(\alpha _{x}(L|_{B})\geqslant \gamma /d\) then \(\alpha _{x}(L)\geqslant \gamma /d\).

     
Furthermore, there is a subvariety Y of X such that \(x\in Y\) and for all \(y\in X(k)\), we have Open image in new window, provided that Open image in new window is in the effective cone of Y.

Proof

Let Open image in new window be a sequence approximating x. If infinitely many \(x_i\) lie outside of B then we may pass to the subsequence of points outside of B, which can only approximate the point x better than the sequence as a whole. To prove part (a) we may therefore assume that all points of \(\{x_i\}\) lie outside B. Base changing \(\{x_i\}\) we obtain a sequence \(\{y_i\}\) in \(X_K(K)\) approximating \(x \in X_K(K)\). Applying Lemma 3.2 to \(X_{K}\), we conclude that Open image in new window and \(\alpha (\{y_i\},L)_{K} \geqslant \gamma \). Since there is a sequence of k-points approximating x we conclude by Remark 2.7 that (in the notation of Proposition 2.6) \(m_v=1\). Therefore by Proposition 2.6, Open image in new window and \(\alpha (\{x_i\},L)_{k} = \alpha (\{y_i\},L)_{K}/d\geqslant \gamma /d\), proving (a).

If \(\alpha _{x}(L)< \gamma /d\) then there must be a sequence \(\{x_i\}\) approximating x such that \(\alpha (\{x_i\},L)< \gamma /d\). By part (a), this implies that all but finitely many \(x_i\) lie in B. Thus \(x\in B\) since B is closed. Since omitting finitely many elements of a sequence does not change the approximation constant we may assume that all \(x_i\) are contained in B. Since \(\alpha _{x,X}(L)\) is the infimum of the approximation constants for sequences \(\{x_i\}\) with \(\alpha (\{x_i\},L)< \gamma /d\) we conclude that \(\alpha _{x}(L) = \alpha _{x}(L|_{B})\) proving (b).

If \(\alpha _{x}(L) < \gamma /d\) then part (b) along with the hypothesis for part (c) lead to an immediate contradiction. Thus, under the hypotheses of part (c), \(\alpha _{x}(L)\geqslant \gamma /d\).

The final remark is obvious in cases (a) and (c), and in case (b), we may replace X with B and L with \(L|_B\) and apply Theorem 3.3 again. Iterating this, we deduce the desired result. \(\square \)

Remark 3.4

Theorem 3.3 still holds if we replace B by the Zariski closure of B(k). This has the added advantage that every component of B is then absolutely irreducible (see [10, Lemma 2.17]).

Corollary 3.5

Suppose that L is a nef line bundle on X such that Open image in new window has empty stable base locus for all rational \(\gamma \in (0,\epsilon _x(L))\). Then \(\alpha _{x}(L) \geqslant \epsilon _{x}(L)/d\).

Proof

For all rational \(\gamma \in (0,\epsilon _x(L))\) the stable base locus of Open image in new window is empty by hypothesis, and thus, by Theorem 3.3 (a), we conclude that \(\alpha _{x}(L)\geqslant \gamma /d\) for any such \(\gamma \), and hence that \(\alpha _{x}(L)\geqslant \epsilon _{x}(L)/d\). \(\square \)

Corollary 3.6

If L is an ample line bundle, then \(\alpha _x(L)\geqslant \epsilon _x(L)/d\).

Proof

For all rational \(\gamma \in (0,\epsilon _x(L))\) the line bundle \(\pi ^{*}L-\gamma E\) is ample on \(\widetilde{X}\). (See the discussion after Definition 2.9 for a proof of this.) In particular, the stable base locus of Open image in new window is empty, so the result follows by Corollary 3.5. \(\square \)

Remark 3.7

If \(X=\mathbb {P}^1\) then Corollary 3.6 and the fact that \(\epsilon _{x}({\mathscr {O}}_{\mathbb {P}^n}(1))=1\) give \(\alpha _{x}({\mathscr {O}}_{\mathbb {P}^1}(1))\geqslant 1/d\). Thus on \(\mathbb {P}^1\) Corollary 3.6 amounts to the classic Liouville bound \(\tau _x \leqslant d\). Liouville’s original result also includes an explicit relation between the height and distance of approximating rational points, and Theorem 3.3 has this feature as well. For this reason we consider Theorem 3.3 and its corollaries to be “Liouville bounds” for \(\alpha _x\).

The effective cone is usually larger than the nef cone, and in general the parts of Theorem 3.3 imply a much stronger lower bound for \(\alpha _{x}(L)\) than Corollary 3.5. We will use this in the next section to compute \(\alpha \) for the cubic surface, but give a brief illustration now by calculating \(\alpha \) for rational points of a non-split quadric surface in Open image in new window. (A split quadric surface is isomorphic to Open image in new window, and Open image in new window when \(a,b>0\), as implied by Proposition 2.11 (d) and computed in both [9, Theorem 3.1] and [10, Section 2; Example (b) just before Lemma 2.17].)

Example 3.8

Let X be a smooth quadric surface in Open image in new window defined over k, and set \(L={\mathscr {O}}_{\mathbb {P}^3}(1)|_{X}\). We assume that no lines on X are defined over k. Let x be a k-point of X. By intersecting with a (rationally defined) hyperplane we may find a conic C passing through x such that C is isomorphic to \(\mathbb {P}^1\) over k. By Lemma 2.5 and Proposition 2.11 (a, c), we therefore have \(\alpha _{x,X}(L) \leqslant \alpha _{x,C}(L|_{C}) = \alpha _{x,\mathbb {P}^1}({\mathscr {O}}_{\mathbb {P}^1}(2)) = 2\). Since x lies on a line (over \(\overline{k}\)), we have \(\epsilon _{x}(L)=1\), and, applying Corollary 3.5, we obtain \(\alpha _{x}(L)\geqslant 1\). Thus \(1\leqslant \alpha _{x}(L) \leqslant 2\), i.e., Corollary 3.5 does not give enough information to determine \(\alpha _{x}(L)\) in this case.

However, let \(\pi :\widetilde{X}\rightarrow X\) be the blowup of X at x with exceptional divisor E. Then Open image in new window is effective with base locus the proper transform of the two lines passing through x. In particular the image B of this base locus is the union of the two lines of ruling passing through x. Since (by assumption) neither of these lines is defined over k, x is the only k-point of B. Thus, by Theorem 3.3 (a), if \(\{x_i\}\) is any sequence of k-points approximating x then \(\alpha (\{x_i\},L)\geqslant 2\), and in particular \(\alpha _{x}(L)\geqslant 2\). Thus \(\alpha _{x}(L)=2\) for all k-points of X.

Since X is non-split the Picard group of X (over k) has rank one with generator L. Thus the above computation and the homogeneity in Proposition 2.11 (a) determines \(\alpha \) for all \(x\in X(k)\) and all ample line bundles on X defined over k.

4 The cubic surface

In this section, we will compute \(\alpha _x\) and \(\epsilon _x\) for all k-rational points x on the blowup X of \(\mathbb {P}^2\) at six k-rational points in general position. To begin, we will recall some notions from [9].

Definition 4.1

A sequence \(\{x_i\}\rightarrow x\) whose approximation constant is equal to \(\alpha _x(L)\) (if such a sequence exists) is called a sequence of best approximation to x. A curve C passing through x is a called a curve of best approximation (with respect to L) if C contains a sequence of best approximation to x.

In other words, if C is a curve of best approximation to x on X, then the rational points on C approximate x roughly as well as the rational points on X approximate x.

Note also that for a point x on a curve C and an ample divisor L on C, there is always a sequence Open image in new window such that \(\alpha _x(\{x_i\},L)=\alpha _x(L)\). Thus, in particular, C is a curve of best approximation if and only if \(\alpha _x(L|_C)=\alpha _x(L)\).

In the example of the non-split quadric—and in many others considered in [9]—there is always a curve of best approximation to x. In [9, Section 4], it is shown that if Vojta’s main conjectures (see [13] for statements) are true, then \(\alpha _{x}(L)\) finite implies that \(\alpha _{x}(L)\) is computed on a subvariety \(V\subseteq X\) of negative Kodaira dimension (possibly X itself, if X has negative Kodaira dimension). Since varieties of negative Kodaira dimension are (again, conjecturally) covered by rational curves, one is led to the following further prediction:

Conjecture 4.2

([9, Conjecture 2.7]) Let X be an algebraic variety defined over k, and L any ample divisor on X. Let x be any k-rational point on X and assume that there is a rational curve defined over k passing through x. Then there exists a curve C (necessarily rational) of best approximation to x on X with respect to L.

In [9], the first author proves this conjecture in many cases, and shows that in many others it follows from Vojta’s Conjecture. Those proofs use a slightly different definition of \(\alpha \), but the proofs do not essentially change in the new setting.

The Seshadri-constant analogue of a curve of best approximation is called a Seshadri curve (cf. Proposition 2.10):

Definition 4.3

Let L be a nef divisor on an algebraic variety X, and \(x\in X\) any point. A Seshadri curve for x with respect to L is a curve C such that Open image in new window.

In all currently known examples, there exists a Seshadri curve for x with respect to L, but it is conjectured that this is not always the case. In particular, it is possible that the Seshadri constant might sometimes be irrational (see [7, Remark 5.1.13]).

It is useful to know that for a fixed curve C, the set of line bundles for which C is a curve of best approximation form a subcone of the Néron–Severi group, and similarly for the property of being a Seshadri curve.

Proposition 4.4

Let X be a variety over \(\mathrm{Spec}(k)\), and let \(x\in X(k)\) be any k-rational point. Let \(D_1\) and \(D_2\) be nef divisors on X with height functions \(H_1\) and \(H_2\) bounded below by a positive constant in some neighbourhood of x. Let \(a_1\) and \(a_2\) be non-negative integers, and let \(D=a_1D_1+a_2D_2\).
  1. (a)

    If C is a curve of best approximation for \(D_1\) and \(D_2\), then C is also a curve of best approximation for D.

     
  2. (b)

    If C is a Seshadri curve for x with respect to \(D_1\) and \(D_2\), then C is also a Seshadri curve for x with respect to D.

     

Proof

Part (a) appears as [9, Corollary 3.2]. To prove part (b), note that Proposition 2.11 (b) implies the estimateOn the other hand, the hypotheses of part (b) giveThus, by Proposition 2.10, \(a_1\epsilon _{x}(D_1)+a_2\epsilon _{x}(D_2)\) is an upper bound for \(\epsilon _{x}(D)\). Therefore Open image in new window and C is a Seshadri curve for D, proving (b). \(\square \)

We are now ready to begin the proof of the main result of this section. Before we state and prove the general result, we will illustrate the fundamental techniques in the case \(L=-K\).

Theorem 4.5

Let X be a smooth cubic surface in Open image in new window defined over k, and isomorphic over k to the blowup of Open image in new window at six k-rational points in general position. Let \(x\in X(k)\) be any k-rational point, and let \(C_x\) be the curve of intersection of X with the tangent plane to X at x. Then
$$\begin{aligned} \epsilon _{x}(-K) = {\left\{ \begin{array}{ll} \,1 &{} \quad \text{ if } x \text { lies on one of the 27 lines of }X; \\ \,\displaystyle \frac{3}{2} &{}\quad \text{ otherwise, } \end{array}\right. } \end{aligned}$$
while

Proof

Set \(L=-K=\mathscr {O}_{\mathbb {P}^3}(1)|_{X}\), and let x be a point of X(k). If x lies on a line \(\ell \) then, by Proposition 2.11 (c), we have \(\epsilon _{x,\ell }(L|_{\ell }) \geqslant \epsilon _{x,X}(L) \geqslant \epsilon _{x,\mathbb {P}^3}(\mathscr {O}_{\mathbb {P}^3}(1)).\) Since \(\epsilon _{x,\ell }(L|_{\ell })=\epsilon _{x,\mathbb {P}^3}(\mathscr {O}_{\mathbb {P}^3}(1))=1\), we conclude that \(\epsilon _{x}(L)=1\). Similarly (using Proposition 2.11 (c) again and the fact that Open image in new window by Lemma 2.5) we conclude that \(\alpha _{x}(L)=1\).

We now suppose that x does not lie on a line. Let \(\pi :\widetilde{X}\rightarrow X\) be the blowup of X at x, with exceptional divisor E. Then \(C_x\) is a Seshadri curve for x with respect to L. To see this, note first that \(C_x\) satisfies Open image in new window, so \(\epsilon _x(L)\leqslant 3/2\). Conversely, if \(a>3/2\), then Open image in new window is not nef, because Open image in new window and Open image in new window is the class of the proper transform of \(C_x\). Thus, \(\epsilon _x(L)\geqslant 3/2\), and so \(\epsilon _x(L)=3/2\), and \(C_x\) is a Seshadri curve for x with respect to L.

We now turn to the computation of \(\alpha \). The stable base locus of Open image in new window is Open image in new window, the proper transform of \(C_x\). Hence, by Theorem 3.3 (b), either \(\alpha _{x}(L)\geqslant 2\) or \(\alpha _{x}(L) = \alpha _{x,C_x}(L|_{C_x})\) (note that \(d=1\)). By intersecting X with a hyperplane containing x and one of the lines, we produce a k-rational conic passing through x, and approximating on the conic gives us \(2\geqslant \alpha _{x}(L)\). We therefore conclude that Open image in new window.

The curve \(C_x\) is singular at x, and since x does not lie on a line, \(C_x\) is also irreducible. In particular, \(C_x\) is an irreducible curve of geometric genus zero, and since x is defined over k, \(C_x\) is birational to \(\mathbb {P}^1\) over k, via projection from x in the tangent plane.

Applying Theorem 2.8 to \(C_x\), we find thatand this implies the stated values of \(\alpha _{x}(L)\) above. \(\square \)

We now treat the case of a general nef divisor D. In what follows, we assume that the point x does not lie on a \((-1)\)-curve on X. We begin with a calculation of the Seshadri constant \(\epsilon \). To do this, we will need some notation.

Let Open image in new window be the blowing down map, and let \(E_1,\ldots ,E_6\) be the exceptional divisors of \(\varphi \). We define the following linear equivalence classes on X:
  • \(L=\varphi ^*{\mathscr {O}}(1)\),

  • \(L_i=L-E_i\), the strict transform of a line through \(P_i=\varphi (E_i)\),

  • \(L_{ij}=2L-\bigl (\sum E_n\bigr )+E_i+E_j\) (where \(i\ne j\)), the strict transform of a conic through the four points \(P_n\) with \(n\ne i,j\),

  • \(B_i=3L-\bigl (\sum E_n\bigr )-E_i\), the strict transform of a cubic curve through all six points \(P_n\), with a node at \(P_i\).

Let \(h\) be the class of a hyperplane in the anticanonical embedding Open image in new window. For any line \(\ell \) on X, the hyperplanes containing \(\ell \) give (after removing \(\ell \)) a base-point-free pencil on X. If \(x\in X\) does not lie on a line then the unique curve in this pencil through x is smooth and irreducible. The classes Open image in new window defined above are the 27 pencils coming from the lines. Recall that for any point x on X we use \(C_x\) for the intersection of X with its tangent plane at x (so \(C_x\) has class \(h\)). If x does not lie on a line, then \(C_x\) is a plane cubic curve with one double point, at x.

Theorem 4.6

Let x be a point on X that does not lie on a \((-1)\)-curve, and let D be a nef divisor on X. The Seshadri constant \(\epsilon _x(D)\) is equal to Open image in new window.

Proof

The nef cone \(\Gamma \) of X has 99 generators, which are listed in Sect. 5, Table 1. Let S be the set of 27 divisor classes \(\{L_i,L_{ij},B_i\}\) as i and j range over all possible values, and for each element C in S, we define the subcone \(\Gamma (C)\) byFurther define the subcone \(\Gamma (h)\) to beIt is clear that \(\Gamma \) is the union of these 28 subcones. To prove Theorem 4.6, it suffices to show that for every subcone \(\Gamma (C)\), with \(C\in S\), the curve through x in the pencil corresponding to C is a Seshadri curve for x with respect to D for all \(D\in \Gamma (C)\) (respectively, in the case of the subcone \(\Gamma (h)\), that \(C_x\) is a Seshadri curve for x with respect to D for all \(D\in \Gamma (h)\)). By Proposition 4.4 (b), it further suffices to prove this for D a generator of the cone \(\Gamma (C)\) (respectively \(\Gamma (h)\)).

The fundamental group of the space of all smooth cubic surfaces acts via monodromy on the Néron–Severi lattice of X. This monodromy action preserves the hyperplane class h and acts transitively on the classes of the lines. Thus, up to monodromy action, there are only two of these subcones: \(\Gamma (L_1)\) and \(\Gamma (h)\). Generators for each of these subcones can be found in Sect. 5. Let \(F=F_{x,L_1}\) be the unique curve in the pencil \(L_1\) passing through x. For each generator D of \(\Gamma (L_1)\), it is straightforward to verify that F is a Seshadri curve for x with respect to D. These verifications also appear in Sect. 5. Each generator G of \(\Gamma (h)\) is also a generator of one of the other twenty-seven subcones \(\Gamma (C)\), and for each such G, we have Open image in new window. Thus, since C is a Seshadri curve for x with respect to G, it follows that \(C_x\) is also a Seshadri curve for x with respect to G, and so \(C_x\) is a Seshadri curve for every element of \(\Gamma (h)\). \(\square \)

The next step is to calculate \(\alpha _x\) for a point on a cubic surface.

Theorem 4.7

Let \(x\in X(k)\) be a point that does not lie on a \((-1)\)-curve, and let D be a nef divisor on X. If the tangent curve \(C_x\) is a cuspidal cubic, or a nodal cubic whose tangent lines at x are defined over \(k_v\) but not defined over k, then \(\alpha _x(D)=\epsilon _x(D)\). Otherwise, Open image in new window.

Proof

Suppose that D is in one of the cones \(\Gamma (C)\) for \(C\in S\), and let \(F_{x,C}\) be the element of the pencil corresponding to C which passes through x. Since \(F_{x,C}\) is a smooth k-rational curve, we have(Note that the blowup of X at a point not on a line is a del Pezzo surface of degree two, and so every nef divisor on the blowup is semiample (see [1, Theorem 5.1.2.1]), and thus has nonempty stable base locus. In particular the hypotheses of Corollary 3.5 are satisfied.) Thus Open image in new window and \(F_{x,C}\) is a curve of best approximation with respect to D.

Now suppose that \(D\in \Gamma (h)\). If \(C_{x}\) is cuspidal, or nodal with tangent lines having slopes in \(k_v\) but not k, then Theorem 2.8 gives Open image in new window. By Theorem 4.6, Open image in new window, and so as above we conclude that Open image in new window, and that \(C_x\) is a curve of best approximation for D.

We now assume that \(C_{x}\) is nodal and the slopes of the tangent lines are in k or not in \(k_v\). The codimension one faces of \(\Gamma (h)\) (i.e., the facets) occur where one of the inequalities defining \(\Gamma (h)\) becomes an equality, so that each facet is the intersection of \(\Gamma (h)\) and \(\Gamma (C)\) for some \(C\in S\). For each \(C\in S\) set \(\widehat{\Gamma }(C)\) to be the cone generated by \(\Gamma (C)\) and \(-K\). Since \(-K\) is in the interior of \(\Gamma (h)\) it follows that \(\Gamma \) is the union of the \(\widehat{\Gamma }(C)\), \(C\in S\).

As above, for any \(C\in S\), let \(F_{x,C}\) be the member of the pencil corresponding to C passing through x. In the proof of Theorem 4.5 we have seen that \(F_{x,C}\) is a curve of best approximation for \(-K\), and in the first part of the argument above that \(F_{x,C}\) is a curve of best approximation for all \(D\in \Gamma (C)\). By Proposition 4.4 (a), we conclude that \(F_{x,C}\) is a curve of best approximation for all \(D\in \widehat{\Gamma }(C)\). The result follows.\(\square \)

Note that as part of the proof we have shown that Conjecture 4.2 holds for every point \(x\in X\) not on a \((-1)\)-curve.

Footnotes

  1. 1.

    This property appears as the statement “\(F_{p,q}\) is ample whenever \(p>q/\epsilon _x(L)\)” in [4, p. 98].

Notes

Acknowledgments

We thank Kevin Hare for useful computations performed in the initial stages of this project. We are grateful to the referees of this paper who provided many useful suggestions on the exposition and advice on strengthening one of the results. Finally, we wish to acknowledge an intellectual debt to Mike Nakamaye, who has long advocated the point of view that Seshadri constants are diophantine.

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Authors and Affiliations

  1. 1.Department of Pure MathematicsUniversity of WaterlooWaterlooCanada
  2. 2.Department of Mathematics and StatisticsQueens UniversityKingstonCanada

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