Lang–Vojta conjecture over function fields for surfaces dominating ${{\mathbb {G}}}_m^2$

Abstract

We prove the nonsplit case of the Lang–Vojta conjecture over function fields for surfaces of log general type that are ramified covers of ${{\mathbb {G}}}_m^2$. This extends the results of Corvaja and Zannier (J Differ Geom 93(3):355–377, 2013), where the conjecture was proved in the split case, and the results of Corvaja and Zannier (J Algebr Geom 17(2):295–333, 2008), Turchet (Trans Amer Math Soc 369(12):8537–8558, 2017) that were obtained in the case of the complement of a degree four and three component divisor in ${{\mathbb {P}}}^2$. We follow the strategy developed by Corvaja and Zannier and make explicit all the constants involved.

1 Introduction

The celebrated Lang–Vojta conjecture, see [24, Conjecture F.5.3.6], predicts degeneracy of S-integral points on varieties of log general type over number fields. It is known in full generality for curves, where it reduces to Siegel’s theorem (see for example [35]), and for subvarieties of semi-abelian varieties [22, 39, 40]. Very deep results have been obtained applying the method developed by Corvaja and Zannier [15], building on [14], which led to the proof of the conjecture in several new cases, e.g. [5, 12, 17, 19, 29] (see [10, 21] for surveys of known results).

In the case of function fields, the Corvaja and Zannier strategy allows one to obtain results that are still out of reach with the current methods in the number field case: for example, in [18] the authors prove the split case of the conjecture for the complement of a conic and two lines in ${{\mathbb {P}}}^2$, a problem which is still open over number fields. The latter result has then been generalized in [20] for isotrivial surfaces that are ramified covers of ${{\mathbb {G}}}_m^2$ (see also [9, 31, 32] for analogous results in the compact and analytic cases).

The goal of this article is to prove the non-isotrivial case of [20]. The setting is the following: let $\kappa$ be an algebraically closed field of characteristic 0, let $\kappa ({{\mathscr {C}}})$ be the function field of a nonsingular projective curve ${{\mathscr {C}}}$ and let S be a finite set of points of ${{\mathscr {C}}}$. Let (X, D) be a pair of log general type over $\kappa ({{\mathscr {C}}})$, where X is a nonsingular projective surface, D is a simple normal crossings Cartier divisor on X and there is a dominant morphism over $\kappa ({{\mathscr {C}}})$. Let $({{\mathscr {X}}},{{\mathscr {D}}})$ be a log canonical model of (X, D) such that there exists a generically finite dominant morphism .

Theorem B

Let Z be the closure of the ramification divisor of $\pi \!\restriction _{{{\mathscr {X}}}\setminus {{\mathscr {D}}}}$ and assume that the image of the generic fiber  $\pi (Z_\eta )$ is disjoint from the singular points of . Then, for every projective embedding $\varphi$ of ${{\mathscr {X}}}$, there exists an explicit positive constant $C = C(Z_\eta ,\deg \pi , \varphi )$ such that every section $\sigma :{{\mathscr {C}}}\rightarrow {{\mathscr {X}}}$ with satisfies

where $\chi _S({{\mathscr {C}}})$ is the (negative) Euler characteristic of the affine curve , i.e., $\chi _S({{\mathscr {C}}}) = 2 g({{\mathscr {C}}}) - 2 + \#\, S$.

The main application of Theorem A is to complements of normal crossings divisors in ${{\mathbb {P}}}^2_{\kappa ({{\mathscr {C}}})}$.

Theorem B Let D be a divisor in ${{\mathbb {P}}}^2_{\kappa ({{\mathscr {C}}})}$ of degree $d \geqslant 4$ with $r \geqslant 3$ components. Let ${{\mathscr {D}}}$ be the closure of D in and let S be a finite set of points of ${{\mathscr {C}}}$ such that, for every $P \notin S$, the fiber ${{\mathscr {D}}}_P$ has normal crossings singularities. Then, for every projective embedding $\varphi$ of , there exists a constant $C = C(D,\varphi )$ such that every section with verifies

The isotrivial case of Theorem B was proved in [20, Theorem 1], and previously in [18, Theorem 1.1] for $d=4$ and $r=3$; the latter case was obtained in [37, Theorem 1.3] for non-isotrivial pairs. Note that, when $r \geqslant 4$, the conclusion is known to hold essentially by a reduction to [7] or [42].

Remark 1.1

In the statement of Theorem A, given a projective embedding of ${{\mathscr {X}}}$, the degree of $\sigma ({{\mathscr {C}}})$ is bounded by a constant C multiplied by , where the constant C depends only on the geometric data of the finite map $\pi$ and its ramification. The dependence on the curve appears only in the Euler characteristic. In fact the proof shows that C depends only on the generic fiber $(X,D) \rightarrow {{\mathbb {G}}}_m^2$ and the projective embedding. This implies that, given a finite cover ${{\mathscr {E}}}\rightarrow {{\mathscr {C}}}$ and a pair $({{\mathscr {X}}},{{\mathscr {D}}}) \rightarrow {{\mathscr {C}}}$ as before, one will obtain the same result for the pair $({{\mathscr {X}}}, {{\mathscr {D}}}) \times _{{\mathscr {C}}}{{\mathscr {E}}}$ with the same constant C (up to the choice of a compatible embedding). This is consistent with more general conjectures of Vojta (see Sect. 1.1 or [4, Section 10.2] for a more detailed discussion).

The main idea in the proof of Theorem A, as in [20], is to estimate the contribution of the ramification divisor of the finite map $\pi$ to the height of a section $\sigma \in X(\kappa ({{\mathscr {C}}}))$. More precisely, the strategy of the proof is the following: in Sect. 3 we obtain a preliminary result on S-units which satisfy a multiplicative dependence relation; this is used in Sect. 4, where we prove an explicit bound for the number of multiple zeros of polynomials in $\kappa ({{\mathscr {C}}})[X,Y]$ evaluated at S-units, extending [18, Theorem 1.2]. This latter result is the key point to estimate the contribution of the ramification divisor Z to the height of a section, which is obtained in Sect. 5.

In the nonsplit case one needs to deal with the problem that the log general type assumption does not guarantee in general the positivity of the ramification divisor Z. We discuss various results about the divisor Z in Sect. 6. In particular, we show that, even if the divisor Z might not be big, its twist by the pull-back of a positive divisor on ${{\mathscr {C}}}$ is big. Moreover, in the case in which the model of the divisor D is ample, the ramification divisor itself can be shown to be big (see Proposition 6.4).

We prove Theorem A in Sect. 7, where we apply Proposition 5.1 together with the generalized abc inequality over function fields [7, Theorem B]. Lastly, in Sect. 8, as an application of Theorem A, we prove Theorem B and give an explicit example in the case where the base curve is ${{\mathbb {P}}}^1$.

Remark 1.2

One could ask what happens if, instead of considering ramified covers of ${{\mathbb {G}}}_m^2$, one considers ramified covers $\pi :X \rightarrow A$ of an abelian variety A. By a result of Kawamata, there exists a finite étale cover such that $X'$ dominates a variety of general type. Lang’s Conjecture would then predict that, over a number field k, the rational points of X are not Zariski dense and, in our knowledge, this problem is still open even in dimension 2 (also over function fields). On the other hand, it was recently proved in [11] that the set is always Zariski dense when A(k) is dense.

1.1 Connections with Vojta’s conjectures

A celebrated conjecture in diophantine geometry, proposed by Vojta [38, Conjecture 3.4.3], predicts an arithmetic analogue of a (conjectural) higher dimensional Second Main Theorem in Nevanlinna Theory. The conjecture can be extended to describe the distribution of algebraic points on a non-singular projective variety X defined over a number field k and a normal crossings divisor D on X. In the stronger form with truncation the conjecture reads as follows:

Conjecture 1.3

(See [41, Conjecture 24.3]) Let X, D, k be as above. Let S be a finite set of places of k containing the Archimedean ones, let $K_X$ be the canonical divisor of X, let A be an ample divisor on X and let r be a positive integer. Then, for every $\varepsilon > 0$, there exists a proper Zariski closed subset $W = W(k,S,X,D, A,\varepsilon ,r)$ of X such that the inequality

$$\begin{aligned} h_{K_X + D}(x) -\varepsilon h_A(x) \leqslant d_k(x) + N_S^{(1)}(D,x) + O(1) \end{aligned}$$

holds for almost all with $[k(x) \,{:}\, k] \leqslant r$.

In the statement of the conjecture, $d_k(x)$ denotes the logarithmic discriminant of the point x (see [41, Definition 23.1]) and $N_S^{(1)}(D,x)$ denotes the truncated counting function (see [41, Definition 22.4]). Conjecture 1.3 has a wide range of important consequences; we mention for example the abc conjecture of Masser–Oesterlé [33, Conjecture 3], the Bombieri–Lang conjecture [24, Conjecture F.5.2.1] and the Lang–Vojta conjecture [24, Conjecture F.5.3.6]. We refer to [1, 2, 4, 25] for further consequences.

In this paper we deal with the function field case of Conjecture 1.3, i.e., when the number field k is replaced by the function field $\kappa ({{\mathscr {C}}})$ of a non-singular projective curve over an algebraically closed field $\kappa$ of characteristic zero. We point out that Theorem A is related to the function field analogue of Conjecture 1.3. To see this, consider (X, D) as in Conjecture 1.3: points $x \in X(k)$ correspond over function fields to sections $\sigma :{{\mathscr {C}}}\rightarrow {{\mathscr {X}}}$, for a projective model ${{\mathscr {X}}}$ of the variety X. Similarly, the height bound corresponds to a degree bound for the image $\sigma ({{\mathscr {C}}})$, while the contributions of the discriminant and the truncated counting function correspond to the Euler characteristic $2g({{\mathscr {C}}}) - 2 + \#\, S$.

We also note that, in Theorem A the exceptional set W does not appear explicitly. Moreover, our result shows that a degree bound holds also for sections for which a bound as in Conjecture 1.3 does not hold. On the other hand, our theorem contains a constant C that does not appear in Vojta’s formulation. For further discussion on the exceptional set we refer to [34, Section 3].

The reader can find a detailed analysis of the case $X = {{\mathbb {P}}}^\ell$, ${{\mathscr {C}}}= {{\mathbb {P}}}^1$ and $S = \{0, \infty \}$ in [13, Section 5], where the Nevanlinna analogue is also discussed. Moreover in [13] the authors obtain new cases of Conjecture 1.3 for rational points in higher dimensions, adopting a function field version of the method introduced in [30] (which in turn extended [16]). These results can be seen as higher dimensional cases of [20, Theorem 2], therefore we expect that the methods of the present paper can be further generalized to give higher dimensional analogues of Theorem A.

2 Setting and notations

2.1 Function fields

In this paper we will denote by ${{\mathscr {C}}}$ a nonsingular projective curve (integral, separated scheme of finite type of dimension 1) defined over an algebraically closed field $\kappa$ of characteristic zero and by S a finite set of (closed) points of ${{\mathscr {C}}}$. We will denote by ${{\mathscr {O}}}_S$ the ring of S-integers, i.e., the ring of regular functions in the complement of S: its elements are rational functions on ${{\mathscr {C}}}$ with poles contained in S. Similarly, we will denote by ${{\mathscr {O}}}_S^*$ the group of S-units, i.e., the group of invertible elements of ${{\mathscr {O}}}_S$: its elements are rational functions on ${{\mathscr {C}}}$ with both zeros and poles contained in S. If $g({{\mathscr {C}}})$ is the genus of the curve ${{\mathscr {C}}}$, then the (negative) Euler characteristic of the affine curve , denoted by $\chi _S({{\mathscr {C}}})$, is defined as

If there is at least a nonconstant S-unit, then $\#\, S$ is at least 2 hence $\chi _{S}({{\mathscr {C}}})$ will always be non-negative.

For any rational function $a \in \kappa ({{\mathscr {C}}})$ we denote by $H_{{\mathscr {C}}}(a)$, or simply by H(a) when the reference to the curve is clear, the height of a, i.e., its degree as a morphism to ${{\mathbb {P}}}^1$. This is equivalent to the usual definition of Weil Height via valuations as follows: every point $P \in {{\mathscr {C}}}$ induces a discrete valuation of the field $\kappa ({{\mathscr {C}}})$, trivial on $\kappa$, that can be normalized such that its value group is ${{\mathbb {Z}}}$. We denote by ${{\,\mathrm{ord}\,}}_P$ the corresponding valuation on $\kappa ({{\mathscr {C}}})$. Then, the height of a function $a \in \kappa ({{\mathscr {C}}})$ can be expressed as

If ${{\mathscr {E}}}$ is a nonsingular projective curve and ${{\mathscr {E}}}\rightarrow {{\mathscr {C}}}$ is a dominant morphism (corresponding to an inclusion $\kappa ({{\mathscr {C}}}) \subseteq \kappa ({{\mathscr {E}}})$), then the heights of a rational function $a \in \kappa ({{\mathscr {C}}})$, with respect to ${{\mathscr {C}}}$ and ${{\mathscr {E}}}$, verify

For $n\geqslant 2$ and elements $a_1, \ldots , a_n \in \kappa ({{\mathscr {C}}})$, we denote by $H(a_1\,{:}\, \cdots \,{:}\, a_n)$ the projective height

Given a polynomial $F \in \kappa ({{\mathscr {C}}})[X_1,\dots ,X_n]$, the height of F, denoted by $H_{{{\mathscr {C}}}}(F)$, will always be the maximum of the heights of its coefficients. For more details about heights we refer to [41].

An important tool over function fields is the presence of derivations. Following [18], we will fix a differential form on ${{\mathscr {C}}}$ in order to define the “derivative” of a rational function using the following lemma.

Lemma 2.1

([18, Lemma 3.5]) Given a nonsingular projective curve ${{\mathscr {C}}}$ of genus g and a finite set of points $S \subset {{\mathscr {C}}}$, there exist a meromorphic differential form $\omega \in {{\mathscr {C}}}$ and a finite set $T \subset {{\mathscr {C}}}$ such that and, for every $u \in {{\mathscr {O}}}_S^*$, there exists an $(S \,{\cup }\, T)$-integer $\theta _{u} \in {{\mathscr {O}}}_{S \cup T}$ having only simple poles such that

In the rest of the paper the form $\omega$ will be fixed (compatibly for every finite cover ${{\mathscr {E}}}\rightarrow {{\mathscr {C}}}$) and, for $a \in \kappa ({{\mathscr {C}}})$, we will denote by $a'$ the rational function that satisfies . With this notation, the rational function $\theta _u$ appearing in the previous lemma is equal to $u'/u$.

2.2 Surfaces over function fields and fibered threefolds

The main focus of this paper is non-isotrivial surfaces defined over function fields and their models. We recall here the main definitions, fixing notations and terminology.

Definition 2.2

Given a projective variety X of dimension n defined over the function field $\kappa ({{\mathscr {C}}})$, a (proper) model ${{\mathscr {X}}}$ of X over ${{\mathscr {C}}}$ (or over $\kappa ({{\mathscr {C}}})$) is the datum of a proper flat map $\rho :{{\mathscr {X}}}\rightarrow {{\mathscr {C}}}$ and an isomorphism .

From this definition it follows that the model of a surface X over the function field $\kappa ({{\mathscr {C}}})$ is a fibered threefold ${{\mathscr {X}}}\rightarrow {{\mathscr {C}}}$. We note that, in Definition 2.2, the model ${{\mathscr {X}}}$ can be singular. In this paper we will always restrict to the case in which the total space ${{\mathscr {X}}}$ has only mild singularities: in particular, we will consider only models of nonsingular surfaces that are normal.

When dealing with an affine variety Y, we will identify it with a pair (X, D), where X is a projective variety, D is a normal crossings divisor and . Even if this identification is not unique, in this paper we will use the language of pairs, since it is more natural from the geometric point of view. Moreover, when Y is a nonsingular affine surface, one can always consider a canonical choice for (X, D), namely a minimal log resolution. In this latter case, the pair (X, D) can be chosen to be log smooth, i.e., X is nonsingular and D has simple normal crossings singularities.

Definition 2.3

Given a pair (X, D), a model of (X, D) is a model $\rho :{{\mathscr {X}}}\rightarrow {{\mathscr {C}}}$ of the projective variety X over ${{\mathscr {C}}}$ together with a model of D whose total space is a Cartier divisor ${{\mathscr {D}}}$ of ${{\mathscr {X}}}$. We view the model as a family of pairs $({{\mathscr {X}}},{{\mathscr {D}}}) \rightarrow {{\mathscr {C}}}$. Given an integral affine variety Y, a model of Y is a model of the corresponding pair (X, D).

Similarly as before we will restrict to the case in which the model of a log smooth pair (X, D) has only mild singularities: in particular, we will consider models of a log smooth pair that have log canonical singularities. This still implies that the total space of ${{\mathscr {X}}}$ is normal, but takes into account the presence of the divisor D. We refer to [26, Chapter 2] for the precise definition and properties of log canonical singularities.

In the setting of Conjecture 1.3, we are interested in affine surfaces Y of log general type.

Definition 2.4

A quasi-projective variety Y is of log general type if for a log resolution $({\widetilde{Y}},E)$ of Y, the log canonical divisor $K_{{\widetilde{Y}}} + E$ is big. This property is independent of the choice of the log resolution. If Y is affine corresponding to a pair (X, D), we say that (X, D) is of log general type if is of log general type.

We note that being of log general type does not extend naturally to models. Indeed, if $({{\mathscr {X}}},{{\mathscr {D}}}) \rightarrow {{\mathscr {C}}}$ is a model of a pair (X, D) over the function field $\kappa ({{\mathscr {C}}})$, the fact that (X, D) is of log general type does not imply in general that the log canonical divisor of the total space $({{\mathscr {X}}},{{\mathscr {D}}})$ is big.

Finally we define non-isotrivial pairs: these are pairs that cannot be trivialized after a finite base change, i.e., there exists no finite base change ${{\mathscr {E}}}\rightarrow {{\mathscr {C}}}$ such that the base changed pair is isomorphic to a product , for a pair defined over $\kappa$.

In the case in which the fibers of the model $({{\mathscr {X}}},{{\mathscr {D}}})$ have ample log canonical and mild singularities, being non-isotrivial is equivalent to require that the moduli map ${{\mathscr {C}}}\rightarrow {\overline{{{\mathscr {M}}}}}_h$ to the KSBA moduli space of stable pairs ${\overline{{{\mathscr {M}}}}}_h$ is not constant.

2.3 Threefolds dominating

In this article we consider models of non-isotrivial pairs (X, D) over the function field $\kappa ({{\mathscr {C}}})$ where is a ramified cover of ${{\mathbb {G}}}_m^2$. These correspond to fibrations of the form $\rho :({{\mathscr {X}}},{{\mathscr {D}}}) \rightarrow {{\mathscr {C}}}$, together with dominant maps , that restrict to finite maps in the complement of ${{\mathscr {D}}}$.

We note that in practice the threefold ${{\mathscr {X}}}$ will be given by a dominant rational map whose indeterminacy locus is contained in ${{\mathscr {D}}}$. Moreover, the irreducible components of the indeterminacy locus have dimension at most 1, and their images under the map $\rho$ are finite sets of points of ${{\mathscr {C}}}$ (and contained in the set S). Resolving the indeterminacy of $\pi$ gives a threefold ${{\mathscr {X}}}'$ and a morphism that coincides with $\pi$ in the complement of ${{\mathscr {D}}}$. Moreover, every section $\sigma :{{\mathscr {C}}}\rightarrow {{\mathscr {X}}}$ extend to a section $\sigma ':{{\mathscr {C}}}\rightarrow {{\mathscr {X}}}'$. Therefore we can always assume that $\pi$ is a morphism, up to resolving the indeterminacy locus and replacing it with $\pi '$, since this will affect neither the map in the complement of ${{\mathscr {D}}}$ nor the sections that we will consider.

We denote by Z the closure of the ramification divisor of the restriction $\pi \!\restriction _{{{\mathscr {X}}}\setminus {{\mathscr {D}}}}$. We list here the conditions that will be assumed in the rest of the paper.

• (X, D) is a log smooth non-isotrivial pair of log general type over the function field $\kappa ({{\mathscr {C}}})$ such that is a ramified cover of ${{\mathbb {G}}}_m^2$;

• $\rho :({{\mathscr {X}}},{{\mathscr {D}}}) \rightarrow {{\mathscr {C}}}$ is a log canonical model of (X, D), in particular ${{\mathscr {X}}}$ is normal;

• the fibers of ${{\mathscr {D}}}\rightarrow {{\mathscr {C}}}$ have simple normal crossings singularities outside of S;

• the map $\pi$ is compatible with the fibration, i.e., , where is the second projection;

• on every fiber of $\rho$ outside of S, the map $\pi$ restricts to a finite dominant map to ${{\mathbb {G}}}_m^2$ in the complement of ${{\mathscr {D}}}$ and the restriction of the divisor ${{\mathscr {D}}}$ is the pull-back of the boundary divisor ;

• on every fiber of $\rho$ outside of S, the image of the ramification divisor Z of the restriction of $\pi$ avoids the singular points of the boundary of ${{\mathbb {G}}}_m^2$.

The situation is made explicit in the following diagram:

In the case in which the generic fiber (X, D) of $\rho$ is of log general type, given a projective embedding $\varphi$ of ${{\mathscr {X}}}$, one expects the existence of a constant $C = C(X,D,\pi ,\varphi )$ such that for every section one has

where $\deg _\varphi$ denotes the corresponding degree in the projective space where ${{\mathscr {X}}}$ is embedded. Note that different embeddings give rise to different constants but the existence of the bound is independent of the choice of the embedding. For this reason we will drop the explicit dependence on $\varphi$ and we will assume that all bounds depend on the choice of the embedding.

3 Multiplicative dependence between S-units

In order to prove Theorem A we obtain height bounds that yield the degree bound predicted by the conjecture. Recall that, in the setting of Sect. 2.3, a section , or equivalently a section $\sigma :{{\mathscr {C}}}\rightarrow {{\mathscr {X}}}$ such that , corresponds to a D-integral point in ; similarly, a map such that corresponds to an S-unit point $(u,v) \in {{\mathbb {G}}}_m^2({{\mathscr {O}}}_S^*)$, where u, v are rational functions on ${{\mathscr {C}}}$ with zeros and poles contained in S. The heights of u and v and the degree of $\sigma ({{\mathscr {C}}})$ are strictly related, since a bound on the heights of u and v gives a bound on the degree of the image $\sigma ({{\mathscr {C}}})$. On the other hand, a bound on the degree of the image does not guarantee that the heights of u and v are bounded, since, for example u and v might be multiplicatively dependent (we refer to [20, Section 2] for a detailed discussion). We focus on this latter case in this section.

Let $A(X,Y) \in \kappa ({\mathscr {C}})[X,Y]$ be an irreducible polynomial of the form

where . Let $u_1, u_2 \in \kappa ({{\mathscr {C}}})$ be nonzero rational functions and let $B(X,Y)\in \kappa ({{\mathscr {C}}})[X,Y]$ be defined, in terms of $A(X,Y), u_1, u_2$ as

Note that the derivative of $A(u_1,u_2)$ coincides with $B(u_1,u_2)$.

In this section we derive a dependence relation for S-units $u_1$ and $u_2$ assuming they satisfy a relation of the form $(u_1/\alpha )^r(u_2/\beta )^s = \mu$ for some constant $\mu \in \kappa ^\times$ and $\alpha ,\beta$ roots of A and B. In general, we cannot expect $u_1$ and $u_2$ to satisfy the conclusion of [18, Lemma 3.14], i.e., $u_1^r u_2^s = \mu '$, for some ; instead we prove that there exists a fixed S-unit $\gamma$, independent of $u_1$ and $u_2$, such that $u_1^r u_2^s = \gamma$. This will be sufficient for the applications in this paper.

Lemma 3.1

Let A and B as before and let $(\alpha , \beta )$ be a common zero of A and B in $\kappa ({{\mathscr {C}}})$. If $u_1 /\alpha$ and $u_2 /\beta$ satisfy a multiplicative dependence relation of the form

(3.1)

for a suitable pair of nonzero integers $(r,s)\in {{\mathbb {Z}}}^2$ and a constant $\mu \in \kappa ^\times$, then either one between $u_1/\alpha$ and $u_2/\beta$ is constant or there exists $\gamma \in \overline{\kappa (\lambda _{ij})}$ independent of $u_1, u_2$ such that $u_1^r u_2^s = \gamma$.

Proof

Assume that $u_1/\alpha$ and $u_2/\beta$ satisfy a multiplicative dependence relation of the form (3.1). If $(\alpha , \beta )$ is a singular point of A(X, Y), then it is defined in $\overline{\kappa (\lambda _{ij})}$ independently of $u_1$ and $u_2$; so we can conclude with $\gamma = \mu \alpha ^r \beta ^s$. Therefore, we can assume that at least one between $\frac{\partial }{\partial X} A(\alpha , \beta )$ and $\frac{\partial }{\partial Y} A(\alpha , \beta )$ is non-zero.

Notice moreover that, without loss of generality, we can assume that r, s are coprime integers. If they are not, then we can write $r=r' d$ and $s=s' d$ with ; then, (3.1) would imply that there exists $\mu ' \in \kappa$ such that $\bigl ( \frac{u_1}{\alpha } \bigr )^{r'} \bigl ( \frac{u_2}{\beta } \bigr )^{s'}= \mu '$.

Let us define

Since $A(\alpha , \beta )=0$, taking differentials, we obtain

$$\begin{aligned} \alpha ' \frac{\partial }{\partial X} \,A(\alpha , \beta )+ \beta '\, \frac{\partial }{\partial Y} \,A(\alpha , \beta )+ \Gamma (\alpha , \beta )=0. \end{aligned}$$

(3.2)

By definition of $\alpha$ and $\beta$, we have that $B(\alpha , \beta )=0$, i.e.,

$$\begin{aligned} \frac{u_1'}{u_1}\, \alpha \,\frac{\partial }{\partial X}\, A(\alpha , \beta ) + \frac{u_2'}{u_2} \,\beta \, \frac{\partial }{\partial Y}\, A(\alpha , \beta ) + \Gamma (\alpha , \beta )=0. \end{aligned}$$

(3.3)

By (3.1) we have a linear relation of the form

$$\begin{aligned} r \biggl ( \frac{u_1'}{u_1} - \frac{\alpha '}{\alpha } \biggr ) + s \biggl ( \frac{u_2'}{u_2} - \frac{\beta '}{\beta } \biggr )=0. \end{aligned}$$

(3.4)

Taking the difference between (3.3) and (3.2) and multiplying (3.4) by $\alpha \beta$, we obtain the system

(3.5)

From (3.5) it is easy to see that either $u_1'/u_1 = \alpha '/\alpha$ and $u_2'/u_2 = \beta '/\beta$, which implies that both $u_1/\alpha$ and $u_2/\beta$ are constant, or we have

$$\begin{aligned} s\alpha \,\frac{\partial }{\partial X}\, A(\alpha , \beta ) - r\beta \, \frac{\partial }{\partial Y}\, A(\alpha , \beta )=0 . \end{aligned}$$

We define ; as by assumption A(X, Y) is irreducible and $\deg A^* \leqslant \deg A$, we can have either that $({A}^*(X,Y), A(X,Y))=1$ or, there exists a constant $a \in \kappa$ such that ${A}^*(X,Y)=a A(X,Y)$. Let us analyze the two cases separately.

Suppose first that $({A}^*(X,Y), A(X,Y))=1$. We have that $(\alpha , \beta )$ is a common zero of the polynomials ${A}^*(X,Y)$ and A(X, Y). As the two polynomials are coprime, Bezout’s theorem ensures that the number of common solutions is finite and bounded by $(\deg A)^2$. Hence, $\alpha$ and $\beta$ are two rational functions in $\kappa ({{\mathscr {C}}})$ independent of $u_1$ and $u_2$, and (3.1) can be rewritten as

$$\begin{aligned} u_1^r u_2^s= \mu {\alpha }^r {\beta }^s. \end{aligned}$$

Therefore $\gamma = \mu \alpha ^r \beta ^s$ is an algebraic function of the coefficients of A as wanted.

Let us finally see what happens if ${A}^*(X,Y)=a A(X,Y)$ for some $a \in \kappa$. If $a = 0$, i.e., $A^*(X,Y)$ is identically zero, then the polynomial A is of the form $\lambda _{hr\, hs}(X^r Y^s)^h + \lambda _{00}$ for a non-zero integer h. This implies that $\gamma = \mu \alpha ^r \beta ^s$ is independent of $u_1,u_2$ and $u_1^r u_2^s = \gamma$ as wanted.

On the other hand, if $a \ne 0$, one gets $\lambda _{00}=0$ and, for all i, j such that $1 \leqslant i+j \leqslant \deg A$ and $\lambda _{ij}\ne 0$, we have

$$\begin{aligned} si-rj=a. \end{aligned}$$

(3.6)

Moreover, as by assumption A is irreducible, A has both a monomial that contains only X and a monomial that contains only Y, which implies that a is a non-zero integer divisible by rs (as we are assuming r and s coprime), $sa>0$ and $ra<0$.

Let us assume that $a>0$ (the other case is completely symmetric); then, we must have $s>0$ and $r<0$. From (3.6) and the fact that a is divisible by r and s, the polynomial A will be of the form

$$\begin{aligned} A(X,Y)=\sum _{j=0}^{m}\, \lambda _j X^{-r(m-j)} Y^{sj}, \end{aligned}$$

(3.7)

for some $m>0$. As $A(\alpha , \beta )=0$, from (3.7) we have

$$\begin{aligned} \sum _{j=0}\, \lambda _j (\alpha ^r \beta ^s )^j=0, \end{aligned}$$

therefore $\gamma = \mu \alpha ^r \beta ^s$ is independent of $u_1, u_2$, and $u_1^r u_2^s = \gamma$. $\square$

Remark 3.2

We point out that, as $(\alpha , \beta )$ is a common zero of A and B where B depends on $u_1$ and $u_2$ and their derivatives, writing $u_1^r u_2^s= \mu \alpha ^r \beta ^s$ does not directly give the desired conclusion. In the proof of the previous lemma we however obtained that either $u_1/\alpha$ and $u_2/\beta$ are constant, or the quantity $\mu \alpha ^r \beta ^s$ is an algebraic function of the coefficients of A, and so it is independent of $u_1$ and $u_2$. In particular, if $u_1/\alpha$ and $u_2/\beta$ are not constant, given r, s and $\mu$ there are only finitely many $\gamma \in \kappa ({\mathscr {E}})$ such that $u_1^r u_2^s=\gamma$.

4 Counting multiple zeros

The goal of this section is to prove a bound for the number of multiple zeros of polynomials evaluated at S-units; this extends explicitly [18, Theorem 1.2] to polynomials with nonconstant coefficients. Moreover, we give an explicit bound on the exponents of a multiplicative relation between the S-units, when the bound on the number of multiple zeros might not hold.

Let ${{\mathscr {C}}}$ be a smooth projective curve and let $S \subset {{\mathscr {C}}}$ be a finite set of points. Let $A(X,Y) \in \kappa ({{\mathscr {C}}})[X,Y]$ be a polynomial without repeated factors. We write the polynomial as

where $\deg A = \deg _X A + \deg _Y A$ as before. Recall that we denote by $H_{{{\mathscr {C}}}}(A)$ the height of the polynomial A, which is defined as the maximum of the heights of its coefficients. Then, we have the following result.

Theorem 4.1

Let $\varepsilon > 0$ be a positive real number. Then, there exist constants and such that, for all pairs $(u_1,u_2) \in ({{\mathscr {O}}}_S^*)^2$ with

at least one of the following holds:

• the S-units $u_1,u_2$ verify a relation of the form $u_1^r u_2^s = \gamma$ for a pair of integers such that and $\gamma$ is an algebraic function of the coefficients of A independent of $u_1$ and $u_2$;

• the rational function $A(u_1,u_2)$ verifies

  

  (4.1)

Proof

We factor A(X, Y) in irreducible polynomials in $\kappa ({{\mathscr {C}}})[X,Y]$ as

$$\begin{aligned} A(X,Y) = A_1(X,Y)A_2(X,Y)\cdots A_l(X,Y). \end{aligned}$$

(4.2)

We begin by noticing that, for every $i=\, \ldots , l$, we can enlarge the set S to a set $S_i$ so that all the coefficients of $A_i$ are S-units. Moreover, the cardinality of the set $S_i$ is bounded by $\# \,S + 2(\deg A_i +1)^2 H_{{{\mathscr {C}}}}(A_i)$. In particular, we can assume that we have extended the set S so that every coefficient of a factor of A is a S-unit, and the new set S has cardinality bounded in terms of $\deg A$ and $H_{{{\mathscr {C}}}}(A)$ (since the height of a factor $A_i$ is bounded in terms of the height of A; see e.g. [6, Theorem 1.7.2]). Therefore, from now on, we will assume that, for every i, $A_i(X,Y) \in {{\mathscr {O}}}_S^*[X,Y]$.

We want to prove a bound for the number of multiple zeros of $A(u_1,u_2)$ in terms of $H_{{{\mathscr {C}}}}(u_1)$ and $H_{{{\mathscr {C}}}}(u_2)$. This is equivalent to bound the multiple zeros of every irreducible factor $A_i(u_1, u_2)$ together with a bound on the number of common zeros of $A_i(u_1, u_2)$ and $A_j(u_1, u_2)$ for every pair $1 \leqslant i < j \leqslant l$.

4.1 Number of multiple zeros of an irreducible polynomial

We begin by proving the bound for the number of multiple zeros of an irreducible polynomial, which is the content of the following proposition.

Proposition 4.2

In the same setting as above, let $A(X,Y) \in {{\mathscr {O}}}_S^*[X,Y]$ be an irreducible polynomial and let $\varepsilon > 0$ be a positive real number. Then, there exist constants ${C_1}= {C_1}(\deg A, H_{{{\mathscr {C}}}}(A),$ $\varepsilon )$ and such that, for every pair $(u_1,u_2) \in ({{\mathscr {O}}}_S^*)^2$ with

at least one of the following holds:

• the S-units $u_1,u_2$ verify a relation of the form $u_1^r u_2^s = \gamma$ for a pair of integers such that and $\gamma$ is an algebraic function of the coefficients of A independent of $u_1, u_2$;

• the rational function $A(u_1,u_2)$ verifies

  

  (4.3)

Proof

Note that, if A(X, Y) does not depend on X or on Y, a bound of the desired form is immediate. Therefore we will assume that A depends nontrivally on both X and Y.

Let us consider the polynomial B(X, Y) given by

(4.4)

then, we have that $A'(u_1, u_2)=B(u_1, u_2)$. We can enlarge S to a set $S'$ including the set T defined in Lemma 2.1 so that all the coefficients of B have no poles outside $S'$, i.e., $B(X,Y) \in {{\mathscr {O}}}_{S'}[X,Y]$ (notice the presence of the derivatives of the coefficients $\lambda _{ij}$ in the expression of B(X, Y)). Using Lemma 2.1, we can bound the cardinality of $S'$ by

(4.5)

and so

(4.6)

We rewrite (4.3) as

We will estimate the two sums separately, dividing the proof into several steps similarly to [18]. Moreover, we will explicitly compute all the constants involved at every step showing that they depend only on $\deg A$, $H_{{{\mathscr {C}}}}(A)$ and $\varepsilon$ as wanted.

Step 1. Either the two polynomials A(X, Y) and B(X, Y) are coprime, or $u_1$ and $u_2$ satisfy a relation of the form $u_1^r u_2^s = \gamma$, with $\gamma$ an algebraic function of the coefficients of A and .

Proof

First, notice that by assumption A is irreducible and neither $\partial A/\partial X$ nor $\partial A/\partial Y$ is identically zero, hence A(X, Y) has at least two monomials. Suppose that the two polynomials A and B are not coprime. Since $\deg B(X,Y)\leqslant \deg A(X,Y)$, there exists $a\in \kappa ({{\mathscr {C}}})^\times$ such that $B(X,Y)=aA(X,Y)$. Using (4.4), we have that $i \frac{u_1'}{u_1}+j \frac{u_2'}{u_2}+ \frac{\lambda _{ij}'}{\lambda _{ij}}\!\!=\!\!a$ for every i, j with $\lambda _{ij}\ne 0$. Let us consider two monomials with $(i,j)\ne (h,k)$; we have

hence

which gives a relation of multiplicative dependence between $u_1$ and $u_2$, i.e.,

$$\begin{aligned} u_1^{i-h} u_2^{j-k}= \mu \lambda _{hk} \lambda _{ij}^{-1}, \end{aligned}$$

where $\mu \in \kappa ^\times$. Taking $r=i-h$, $s=j-k$ and $\gamma =\mu \lambda _{hk} \lambda _{ij}^{-1}$, we have a relation $u_1^ru_2^s=\gamma$ with $\gamma$ an algebraic function of the coefficients of A and . This proves Step 1. $\square$

From now on, we will therefore assume that A and B are coprime. As $A(u_1,u_2)' = B(u_1,u_2)$, we have that, for every $P \not \in S'$,

(4.7)

since both $A(u_1, u_2)$ and $B(u_1, u_2)$ are $S'$-integers and so the term of the right-hand side is nonnegative.

Let and be the two resultants of A and B with respect to Y and X. If A and B are coprime, then F and G do not vanish identically. Moreover, since F(X) and G(Y) are linear combinations of A(X, Y) and B(X, Y) over ${{\mathscr {O}}}_{S'}[X,Y]$, we have that for every $P \not \in S'$,

(4.8)

It is then enough to bound the gcd of $F(u_1)$ and $G(u_2)$. In order to do this, we first need to prove a bound for the degrees of F and G and for the heights of their coefficients. This is the content of Step 2.

Step 2. There exist positive constants ${C_3}$ and $C_4$ such that the heights of B(X, Y), F(X) and G(Y) are bounded by and the degrees of B(X, Y), F(X) and G(Y) are bounded by $C_4$.

Proof

Given the expression for B(X, Y), we can bound the height of each of its coefficients as

$$\begin{aligned} H_{{{\mathscr {C}}}}\biggl (\frac{u_1'}{u_1}\biggr ) + H_{{{\mathscr {C}}}}\biggl (\frac{u_2'}{u_2}\biggr ) + H_{{{\mathscr {C}}}}\biggl (\frac{\lambda _{ij}'}{\lambda _{ij}}\biggr ) + H_{{{\mathscr {C}}}}(\lambda _{ij}) \leqslant 3 \chi _{S'}({{\mathscr {C}}}) + H_{{{\mathscr {C}}}}(A), \end{aligned}$$

since the first three terms are bounded by the Euler characteristic of using Lemma 2.1. Consequently, . Moreover, by definition, F(X) and G(Y) can be expressed as the determinant of a matrix whose entries are the monomials appearing in A(X, Y) and B(X, Y), and with $N=\deg _Y A(X,Y) + \deg _Y B(X,Y)$ for F and $N=\deg _X A(X,Y)+ \deg _X B(X,Y)$ for G respectively. Using that $\deg _Y B(X,Y)\leqslant \deg _Y A(X,Y)$ and $\deg _X B(X,Y)\leqslant \deg _X A(X,Y)$, we have that the heights of F and G are bounded as follows:

Using (4.6), the first estimate is proved by taking

Finally, since the degrees of F(X) and G(Y) are bounded by , and

we have that gives the desired estimate. $\square$

Next, we need to factor F(X) and G(Y); this can be done in a suitable finite extension of $\kappa ({{\mathscr {C}}})$. However, in order to bound the gcd of $F(u_1)$ and $G(u_2)$, we have to estimate the degree of this extension and the height of the roots of F(X) and G(Y). This can be done as follows.

Step 3. There exist a cover $p:{{\mathscr {E}}}\rightarrow {{\mathscr {C}}}$ of degree bounded by $2C_4$ and a finite set $U \subset {{\mathscr {E}}}$ such that F(X) and G(Y) splits over $\kappa ({{\mathscr {E}}})$ into linear factors and their roots are U-units. Moreover, there exist two constants ${C_5}, {C_6}>0$ such that

(4.9)

Proof

If the polynomials F(X) and G(Y) split over $\kappa ({{\mathscr {C}}})$, then the conclusion holds trivially, so we will assume that this is not the case.

Let us define $\kappa ({{\mathscr {E}}})$ to be the splitting field of F(Z)G(Z) and let us denote by $p:{{\mathscr {E}}}\rightarrow {{\mathscr {C}}}$ the cover corresponding to the field extension. Then, $\deg p \leqslant \deg F + \deg G \leqslant 2 C_4$. To estimate the Euler characteristic of ${{\mathscr {E}}}$ we can use the Riemann–Hurwitz formula. First, note that the ramification of p can arise only over zeros or poles of the discriminants of the irreducible factors of F and G. By construction, the poles are contained in the set $S'$; on the other hand, the number of zeros is bounded by the heights of the discriminants. Recall that the discriminant of a degree n polynomial h is an homogeneous polynomial in the coefficients of h of degree $2n-2$. Therefore, its height is bounded by $2 H(h) \deg h$. In our case, the heights of F(X) and G(Y) are both bounded by as proved in Step 2, and therefore the total number of zeros is bounded by . This, together with (4.5), implies that the cardinality of the support of the ramification divisor is bounded by .

Since each ramification index is at most $C_4$ (as $\deg p \leqslant 2C_4$), the total ramification of p is bounded by . Applying the Riemann–Hurwitz formula, we get

where .

Let us define the set $U'$ as the set of zeros of the constant and leading terms of both F and G and . Note that the cardinality of $U'$ is bounded by , and therefore the cardinality of $U''$ is bounded by . We define . Notice that, by construction, the roots of F and G are U-units in $\kappa ({{\mathscr {E}}})$; this follows from the fact that the coefficients of F and G are $S'$-integers and the product of all the roots of both F and G is a $U''$-unit. The cardinality of U is bounded by $(\deg p) (\# \,U'')$, i.e.,

with , which completes the proof. $\square$

Recall that, in the proof of the previous step, defined the set $U''$ as $U' \cup S'$, where $U'$ is the set of zeros of the constant and leading terms of both F and G, and the set U as $p^{-1}(U'')$, where $p:{{\mathscr {E}}}\rightarrow {{\mathscr {C}}}$ is the cover defined in Step 3.

Using that , we can split the sum in (4.3) as

By (4.7) and the fact that $p(U)=U''$, we can bound the first sum as

hence we get

(4.10)

We now want to bound the first sum of the right-hand side of (4.10) using the polynomials F and G. Since $\kappa ({{\mathscr {E}}})$ is the splitting field of F(Z)G(Z), we can rewrite the two polynomials in $\kappa ({{\mathscr {E}}})$ as

$$\begin{aligned} F(X) = \eta (X - \alpha _1)\cdots (X-\alpha _m) \quad \text{ and } \quad G(Y) = \nu (Y - \beta _1) \cdots (Y - \beta _n), \end{aligned}$$

where $\eta ,\alpha _1,\dots ,\alpha _m$ and $\nu ,\beta _1,\dots ,\beta _n$ are U-units in $\kappa ({{\mathscr {E}}})$ as proved in the previous step.

For every point we have that

using the general property that ; hence, by (4.8), for every we have

(4.11)

Let us now define the set ${{\mathscr {Z}}}= \{ \alpha ,\beta \in \kappa ({{\mathscr {E}}})\,{:}\, A(\alpha ,\beta ) = 0 = B(\alpha ,\beta ),\alpha \beta \ne 0 \}$. This set is in principle strictly smaller than the set of pairs $(\alpha _i,\beta _j)$ such that $F(\alpha _i) = G(\beta _j) = 0$ that we consider in the sum; however, we will show that, eventually replacing U with a bigger set, the inequality (4.11) remains true if we restrict the sum to the the pairs $(\alpha _i, \beta _j) \in {{\mathscr {Z}}}$. This is the content of the following step.

Step 4. There exist a finite set $V \supseteq U$ and a constant $C_V>0$ such that and, for every ,

(4.12)

Proof

Let V be the subset of ${{\mathscr {E}}}$ obtained by enlarging U such that, for every i and j, $A(\alpha _i, \beta _j)$ and $B(\alpha _i, \beta _j)$ are V-units, whenever they are not zero. The cardinality of the set V can be bounded as follows: since $\alpha _i$ and $\beta _j$ are U-units which are roots of F and G respectively, their heights are bounded by , which, using Step 2 and Step 3, is bounded by . Moreover, we can always bound the height of $A(\alpha _i, \beta _j)$ and $B(\alpha _i, \beta _j)$ in terms of the degree of A and the maximum of the heights of their monomials; using that $H_{{{\mathscr {C}}}}(A)\leqslant {C_3}$, this implies that

and similarly, since $\deg B \leqslant \deg A$ and ,

Taking into account that the number of pairs $(\alpha _i, \beta _j)$ is bounded by $(\deg F)(\deg G)$, which is bounded by $C_4^2$ by Step 2, the former two inequalities give

(4.13)

with .

Let us fix . To prove (4.12) we can assume without loss of generality that , otherwise the inequality is trivial since every term of the sum in the right-hand side of (4.12) is nonnegative. In particular, we can assume that ${{\,\mathrm{ord}\,}}_P(A(u_1,u_2)) > 0$. We will show that, if $(\alpha _i, \beta _j) \notin {{\mathscr {Z}}}$, then

$$\begin{aligned} \min \, \{ {{\,\mathrm{ord}\,}}_P(u_1 - \alpha _i), {{\,\mathrm{ord}\,}}_P(u_2 - \beta _j) \} = 0, \end{aligned}$$

hence proving the claim. To see this, consider $(\alpha _i,\beta _j) \notin {{\mathscr {Z}}}$, so for example $A(\alpha _i,\beta _j) \ne 0$. By definition of V, $A(\alpha _i, \beta _j)$ is a V-unit, which implies that ${{\,\mathrm{ord}\,}}_P(A(\alpha _i, \beta _j))=0$ since $P \notin V$. But this implies that $\min \{ {{\,\mathrm{ord}\,}}_P(u_1 - \alpha _i), {{\,\mathrm{ord}\,}}_P(u_2 - \beta _j) \}=0$, otherwise both the differences $u_1-\alpha _i$ and $u_2-\beta _j$ would have a zero in P, and therefore since $A(u_1, u_2)$ has a zero in P, also $A(\alpha _i, \beta _j)$ would have a zero in P, which contradicts the fact that ${{\,\mathrm{ord}\,}}_P(A(\alpha _i, \beta _j))=0$. $\square$

Using the bound for $\chi ({{\mathscr {E}}})$ obtained in (4.9) we get

(4.14)

By (4.12), Step 2, and inverting the order of summation, we have

(4.15)

Let us define $(\alpha _{{\hat{\imath }}}, \beta _{{\hat{\jmath }}})$ as a pair in ${{\mathscr {Z}}}$ for which the maximum is obtained.

In order to estimate the right-hand side of (4.15), we will apply the gcd result of [18] to the V-units $u_1/\alpha _{{\hat{\imath }}}$ and $u_2/\beta _{{\hat{\jmath }}}$. We distinguish two cases according to whether the V-units are multiplicative independent modulo constants or not.

Step 5. Assume that $u_1/\alpha _{{\hat{\imath }}}$ and $u_2/\beta _{{\hat{\jmath }}}$ are multiplicatively independent modulo constants. There exists a constant ${C_7}>0$ such that, if , then

Proof

Since we are assuming that the V-units $u_1/\alpha _{{\hat{\imath }}}$ and $u_2/\beta _{{\hat{\jmath }}}$ are multiplicative independent modulo constants, we get that [18, Corollary 2.3] implies that

(4.16)

Using the bound of $\chi _V({{\mathscr {E}}})$ in (4.14), we can rewrite (4.15) using (4.16) as

(4.17)

To prove the statement we want to relate the heights of $u_1/\alpha _{{\hat{\imath }}}$ and $u_2/\beta _{{\hat{\jmath }}}$ with the heights of $u_1$ and $u_2$. Since $\alpha _{{\hat{\imath }}}$ and $\beta _{{\hat{\jmath }}}$ are roots of the polynomials F and G, by Step 2 we obtain that

Therefore, if

(4.18)

we have that

(4.19)

and similarly for $u_2/\beta _{{\hat{\jmath }}}$. Notice that (4.18) is equivalent to

under this assumption, using (4.19), we can rewrite (4.17) as

where we set . $\square$

In the next step, we deal with the case in which the V-units $u_1/\alpha _{{\hat{\imath }}}$ and $u_2/\beta _{{\hat{\jmath }}}$ are multiplicatively dependent modulo constants. We observe that, as by construction $(\alpha _{{\hat{\imath }}}, \beta _{{\hat{\jmath }}}) \in {{\mathscr {Z}}}$, we can apply Lemma 3.1 which implies that in this case either $u_1$ and $u_2$ satisfy a relation of the form $u_1^r u_2^s = \gamma$, where $\gamma$ is an algebraic function of the coefficients of A, or at least one between $u_1/\alpha _{{\hat{\imath }}}$ and $u_2/\beta _{{\hat{\jmath }}}$ is constant. However, using the bounds for $H_{{{\mathscr {C}}}}(F)$ and $H_{{{\mathscr {C}}}}(G)$ given in Step 2, we have that, if , we can always assume to be in the case in which none of the quotients $u_1/\alpha _{{\hat{\imath }}}$ and $u_2/\beta _{{\hat{\jmath }}}$ is constant.

Step 6. Assume that $u_1/\alpha _{{\hat{\imath }}}, u_2/\beta _{{\hat{\jmath }}}$ are multiplicatively dependent and let$(u_1/\alpha _{{\hat{\imath }}})^r (u_2/\beta _{{\hat{\jmath }}})^s=\mu$ be a generating relation, for some $\mu \in \kappa ^{\times }$. Then, if, we have that $u_1^r u_2^s = \gamma$, where $\gamma$ is an algebraic function of the coefficients of A, and

Proof

Under the dependence assumption, [18, Corollary 2.3] applied to the V-units $u_1/\alpha _{{\hat{\imath }}}$ and $u_2/\beta _{{\hat{\jmath }}}$ gives

(4.20)

As in the previous step, assuming that , we can rewrite (4.15), using (4.20), in terms of the heights of $u_1$ and $u_2$ as

as wanted. Moreover, the relation $u_1^r u_2^s=\gamma$, with $\gamma$ an algebraic function of the coefficients of A, follows applying Lemma 3.1 and using the assumption that . $\square$

Recall that our goal is to prove a bound on the number of multiple zeros of $A(u_1,u_2)$. This problem has been reduced to estimate the following sum:

(4.21)

since by construction $U=p^{-1}(U'')$ where $p:{{\mathscr {E}}}\rightarrow {{\mathscr {C}}}$ is the cover defined in Step 3.

In Steps 5 and 6 we proved an estimate for the first sum appearing in the right-hand side. We are left with providing an estimate for the remaining sum, which we deal with in the following step.

Step 7. There exists a constant ${C_{8}}>0$ such that

Proof

Let us write the polynomial $A(u_1,u_2) = \theta _1 + \cdots +\theta _M$ as a sum of monomials, where no subsum vanishes. All the $\theta _i$ are S-units, in particular they are p(V)-units as $p(V) \supseteq S$; hence,

(4.22)

On the other hand [18, Lemma 3.11], which follows from [43, Theorem 1], implies that

(4.23)

so, combining (4.22) and (4.23), we have

(4.24)

The number of monomials of $A(u_1,u_2)$ is bounded by $(\deg A + 1)^2$, and $\chi _{p(V)}({{\mathscr {C}}}) \leqslant 2g({{\mathscr {C}}}) -2 + \# \,V$. Hence, using the estimate for $\#\,V$ obtained in (4.13), we have

Therefore, (4.24) can be rewritten as

Taking , we have the desired estimate. $\square$

End of the proof of Proposition 4.2

Fix $\varepsilon > 0$. From (4.21), we have that, either $u_1^r u_2^s = \gamma$ with $\gamma$ an algebraic function of the coefficients of A and , or

In the case in which $u_1/\alpha _{{\hat{\imath }}}$ and $u_2/\beta _{{\hat{\jmath }}}$ are independent modulo constants, by Step 5, if , we get

Therefore in this case we obtain that, if with

then

as wanted.

On the other hand, in the case in which $u_1/\alpha _{{\hat{\imath }}}$ and $u_2/\beta _{{\hat{\jmath }}}$ are dependent, by Step 6 we obtain that, if , then

This implies that, either $u_1^ru_2^s=\gamma$ with $\gamma$ an algebraic function of the coefficients of A and , or, if with

then

The two cases imply that, either $u_1^r u_2^s = \gamma$ with where

or, if , with , we get

finishing the proof. $\square$

4.2 Number of common zeros of two irreducible polynomials

To conclude the proof of Theorem 4.1 we need to prove a bound for the number of common zeros of two irreducible polynomials, which is the content of the following proposition.

Proposition 4.3

In the same setting of Theorem 4.1, let $A_1,A_2 \in {{\mathscr {O}}}_S^*[X,Y]$ be two coprime polynomials and let $\varepsilon > 0$ be a positive real number. Then, there exist constants ${D_1}= {D_1}(\deg A_1 , \deg A_2,$ $H_{{{\mathscr {C}}}}(A_1), H_{{{\mathscr {C}}}}(A_2), \varepsilon )$ and such that, for every pair $(u_1,u_2) \in ({{\mathscr {O}}}_S^*)^2$ with

at least one of the following holds:

• the S-units $u_1,u_2$ verify a relation of the form $u_1^r u_2^s = \gamma$ for a pair of integers such that and $\gamma$ is an algebraic function of the coefficients of $A_1$ and $A_2$, independent of $u_1, u_2$;

• the rational functions $A_1(u_1,u_2)$ and $A_2(u_1,u_2)$ verify

  

  (4.25)

Proof

As before, we note that, if one of the two polynomials $A_1,A_2$ does not depend on X or on Y, a bound of the desired form is immediate. Therefore we will assume that both $A_1,A_2$ depend nontrivally on both X and Y. Let us write the polynomials $A_1(X,Y)$ and $A_2(X,Y)$ as

where $\deg A_i(X,Y) = \deg _X A_i(X,Y) + \deg _Y A_i(X,Y)$ for $i=1,2$ as before.

The proof follows the same steps as in the proof of Proposition 4.2, where the polynomial $A_2$ plays the role of the polynomial B. Therefore we will only indicate the required adjustments and compute the corresponding constants. We remark that, as before, all the constants will depend only on the degree and the heights of the polynomials $A_1,A_2$ and $\varepsilon$.

Note that Step 1 is automatically verified since the two polynomials $A_1$ and $A_2$ are coprime.

Let $F(X)=\text{ Res}_Y(A_1(X,Y), A_2(X,Y)) \in {{\mathscr {O}}}_{S}[X]$ and be the two resultants of $A_1$ and $A_2$ with respect to Y and X, which do not vanish identically in view of the coprimality of $A_1$ and $A_2$. As in the previous case, we have that, for every place $P \not \in S$,

(4.26)

It is then enough to bound the gcd of $F(u_1)$ and $G(u_2)$. By definition of F and G we get the following bounds for their degrees and their heights.

Step 2. There exist positive constants ${D_3}$ and ${D_4}$ such that the heights of $A_1(X,Y)$, $A_2(X,Y),$ F(X) and G(Y) are bounded by and the degrees of F(X) and G(Y) are bounded by ${D_4}$.

Proof

The bounds are obtained similarly to Step 2 of Proposition 4.2. In this setting we get

$\square$

In order to bound the gcd of $F(u_1)$ and $G(u_2)$, we have to estimate the degree of the splitting field of F(Z)G(Z) and the height of the roots of F and G.

Step 3. There exist a cover $p:{{\mathscr {E}}}\rightarrow {{\mathscr {C}}}$ of degree bounded by $2{D_4}$ and a finite set $U \subset {{\mathscr {E}}}$ such that F(X) and G(Y) splits over $\kappa ({{\mathscr {E}}})$ into linear factors and their roots are U-units. Moreover, there exist constants ${D_5}, {D_6}>0$ such that

Proof

Letting $\kappa ({{\mathscr {E}}})$ be the splitting field of F(Z)G(Z) and using the same argument as in the previous case, one concludes taking and . $\square$

In $\kappa ({{\mathscr {E}}})$ we can rewrite the two polynomials as

$$\begin{aligned} F(X) = \eta (X - \alpha _1)\cdots (X-\alpha _m) \quad \text{ and } \quad G(Y) = \nu (Y - \beta _1) \cdots (Y - \beta _n), \end{aligned}$$

where $\eta ,\alpha _1,\dots ,\alpha _m$ and $\nu ,\beta _1,\dots ,\beta _n$ are U-units in $\kappa ({{\mathscr {E}}})$ as proved in the previous step. This implies that, for every ,

hence applying (4.26) it follows that

As in the previous case, we want to restrict the sum on the right to the set ${{\mathscr {Z}}}= \{ (\alpha ,\beta ) \in {{\mathscr {E}}}\,{ :}\, A_1(\alpha ,\beta ) = 0 = A_2(\alpha ,\beta ),\, \alpha \beta \ne 0 \}$. In the next step we prove that the same inequality holds when we restrict to points in a finite set $V \supseteq U$ of bounded cardinality.

Step 4. There exist a finite set $V \supseteq U$ and a positive constant $D_V$ such that and, for every ,

(4.27)

Proof

The proof is identical to Step 4 of Proposition 4.2 and we can take

$\square$

We can then split the sum (4.25) as

(4.28)

Using (4.27) and Step 2, we can rewrite the first sum as

(4.29)

Let $(\alpha _{{\hat{\imath }}}, \beta _{{\hat{\jmath }}}) \in {{\mathscr {Z}}}$ be a pair for which the maximum is obtained. We will estimate the right-hand side of (4.29) using the gcd estimate of [18, Corollary 2.3] and relating it to the heights of $u_1$ and $u_2$. We will consider the case when the two V-units $u_1/\alpha _{{\hat{\imath }}}$ and $u_2/\beta _{{\hat{\jmath }}}$ are multiplicatively independent or not modulo constants separately. This is the content of the next two steps.

Step 5. Assume that $u_1/\alpha _{{\hat{\imath }}}$ and $u_2/\beta _{{\hat{\jmath }}}$ are multiplicatively independent modulo constants. Then, there exists a constant ${D_7}>0$ such that, if , then

(4.30)

Proof

We apply [18, Corollary 2.3] to (4.29), noticing that, if , we can bound the heights of $u_1/\alpha _{{\hat{\imath }}}$ and $u_2/\beta _{{\hat{\jmath }}}$ in terms of the heights of $u_1$ and $u_2$. Following the same computation as in the irreducible case, we get that (4.30) holds with

$\square$

We are left with the case in which $u_1/\alpha _{{\hat{\imath }}}$ and $u_2/\beta _{{\hat{\jmath }}}$ are multiplicatively dependent modulo constants, which is the content of the following step.

Step 6. Assume that $u_1/\alpha _{{\hat{\imath }}}, u_2/\beta _{{\hat{\jmath }}}$ are multiplicatively dependent modulo constants and let be a generating relation for some $\mu \in \kappa ^\times$. Then, $u_1^r u_2^s = \gamma$, where $\gamma$ is an algebraic function of the coefficients of $A_1$ and $A_2$ and, if , we have

(4.31)

Proof

As before, we apply [18, Corollary 2.3] to (4.29), and, if , we can bound the heights of $u_1/\alpha _{{\hat{\imath }}}$ and $u_2/\beta _{{\hat{\jmath }}}$ in terms of the heights of $u_1$ and $u_2$. The same computation as in the irreducible case gives (4.31). Notice that since $(\alpha _{{\hat{\imath }}},\beta _{{\hat{\jmath }}})$ is a zero of $A_1$ and $A_2$, this implies directly that $u_1^r u_2^s= \gamma$ where is an algebraic function of the coefficients of $A_1$ and $A_2$. $\square$

In the previous steps we estimated the first sum in the right-hand side of (4.28); in the last step, we bound the second sum.

Step 7. There exists a constant ${D_8}>0$ such that

Proof

The main idea is that the number of common zeros of $A_1(u_1,u_2)$ and $A_2(u_1,u_2)$, counted with multiplicities, is bounded by the number of zeros of $A_1(u_1,u_2)$ counted with multiplicities, and therefore we can reduce the computation to the case of a single polynomial as in the irreducible case. Using the same computation of Step 7 of Proposition 4.2, we can take

$\square$

End of the proof of Proposition 4.3

The end of the proof is analogous to the irreducible case: given $\varepsilon >0$ one considers (4.28) and uses Step 5 and Step 6 to bound the first sum, and Step 7 to bound the second sum. Then, one gets the desired conclusion with the constants

$\square$

End of the proof of Theorem 4.1

Given the factorization of the polynomial A given in (4.2) we have that, either $u_1$ and $u_2$ satisfy a multiplicative dependence relation of the form $u_1^r u_2^s = \gamma$ with and or

We obtain the bound (4.1) with

where $C_1(A_i, \delta )$ is the constant appearing in Proposition 4.2 applied to the polynomial $A_i$ and the real number $\delta$ and $D_1(A_i,A_j,\delta )$ is the constant appearing in Proposition 4.3 applied to the polynomials $A_i$ and $A_j$ and the real number $\delta$. $\square$

Remark 4.4

Recently in [30], Levin obtained a generalization of gcd results over number fields as [8, 16] for polynomials with an arbitrary number of variables. These results have been used to prove some cases of the Lang–Vojta conjectures in higher dimension both over number fields [30] and function fields [13].

5 The ramification divisor

In this section we study the contribution of the ramification divisor of the map $\pi \!\restriction _{{{\mathscr {X}}}\setminus {{\mathscr {D}}}}$ to the height of a section $\sigma :{{\mathscr {C}}}\rightarrow {{\mathscr {X}}}$ with , where ${{\mathscr {X}}}$ is a threefold as in Sect. 2.3 and $S \subset {{\mathscr {C}}}$ is a finite set of points. Recall that we denoted by Z the closure of the ramification divisor of the finite map . The image will be defined by the vanishing of a certain polynomial $A \in \kappa ({{\mathscr {C}}})[X,Y]$. We will use Theorem 4.1 to derive a bound for the degree of the pull-back of the ramification divisor. This is the content of the following proposition.

Proposition 5.1

In the setting above, for every $\varepsilon > 0$ there exists a constant $C = C(\deg A, H_{{{\mathscr {C}}}}(A), \deg \pi , \varepsilon )$ such that, for every section $\sigma :{{\mathscr {C}}}\rightarrow {{\mathscr {X}}}$ of height such that and $\sigma ({{\mathscr {C}}})$ is not contained in Z, the degree of satisfies

Proof

The statement is trivial if $\sigma ({{\mathscr {C}}}) \cap Z$ is empty, and without loss of generality we can assume Z irreducible by applying the same argument to each irreducible component. By construction inside is defined by an irreducible polynomial $A \in \kappa ({{\mathscr {C}}})[X,Y]$, which, since we are assuming Z is not empty, is not a monomial. Furthermore, we can enlarge S so that each coefficient of A is an S-unit (and the extension depends only on $H_{{{\mathscr {C}}}}(A)$ and $\deg A$). The pull-back $\pi ^*A$ is a regular function on . We claim that the degree of is bounded by the sum of the orders of A evaluated at two S-units.

Note that we can assume that Z is a Cartier divisor: it is always ${{\mathbb {Q}}}$-Cartier (see Sect. 6) and therefore there exists a positive integer $\ell$ such that $\ell Z \sim Z'$ with $Z'$ Cartier. Then, since $\sigma ^*(Z) = \frac{1}{\ell } \sigma ^*(Z')$ and $\ell$ is independent of $\sigma$, we can reduce to the case in which Z itself is a Cartier divisor. Therefore, since ${{\mathscr {X}}}$ is a normal variety, given an (affine) open $V \subset {{\mathscr {X}}}$, Z is locally defined in V by an equation $f_V = 0$. The fact that Z is contained in the ramification locus of $\pi$ implies that $f_V^2$ divides $\pi ^*A$ as elements of the local ring ${{\mathscr {O}}}_{V,Z}$. In particular, we can write locally $\pi ^*A = f_V^2 g$ for a regular function g in V. Consider now a point such that $\sigma (P) \in Z \cap V$; the contribution of P to the divisor is . On the other hand, since $\pi ^* A = f_V^2 g$ in ${{\mathscr {O}}}_{V,Z}$ for a regular function g, we have

where for two S-units u, v of ${{\mathscr {C}}}$. Hence, we can bound the degree of by estimating the number of multiple zeros of the polynomial A evaluated at the two S-units u and v. Formally:

(5.1)

Since we are estimating the degree of $\sigma ^*(Z)$ restricted to , the sum on the right includes only . To estimate this sum we want to apply Theorem 4.1. Fix $\varepsilon > 0$ and assume first that u, v are multiplicatively independent, i.e., they do not satisfy any multiplicative relation of the form $u^r v^s = \gamma$ for a suitable pair of nonzero integers and $\gamma$ an algebraic function of the coefficients of A. We can apply Theorem 4.1 to obtain directly from (5.1) a bound of the form

(5.2)

provided that H is bigger than , where $\Theta _1(\deg A, H_{{{\mathscr {C}}}}(A),\varepsilon )$ is the explicit constant of Theorem 4.1. This proves the conclusion in this case.

Assume on the contrary that there exist and a rational function $\gamma \in \kappa ({{\mathscr {C}}})$ which is an algebraic function of the coefficients of A, such that $u^r v^s = \gamma$. Then Theorem 4.1 implies that, either the same conclusion as in (5.2) holds, or there is a bound of the form for a constant $\Theta _2$ that depends only on $\deg A$ and $\varepsilon$. In this latter case, the curve is a curve of degree bounded by $2 \Theta _2 \, H_{{\mathscr {C}}}(\gamma )$. Therefore, the intersection between and $\pi (Z)$ is bounded by $( 2 \Theta _2\, H_{{\mathscr {C}}}(\gamma )) \deg A$.

In this case we obtain that

provided that $H \geqslant {2 \Theta _2 H_{{\mathscr {C}}}(\gamma ) (\deg A) (\deg \pi )}/{\varepsilon }$. Hence setting

gives the desired bound. $\square$

6 Positivity of the ramification divisor

In Proposition 5.1 we estimated the contribution of the ramification divisor Z to the degree of $\sigma ({{\mathscr {C}}})$. In this section we study positivity properties of this divisor. We first show that Z is linearly equivalent to the divisor $K_{{{\mathscr {X}}}/{{\mathscr {C}}}} + {{\mathscr {D}}}$.

Lemma 6.1

(See also [20, Lemma 1] in the split case) Let ${{\mathscr {X}}},{{\mathscr {D}}}, Z$ be defined as in Sect. 2.3. Then

$$\begin{aligned} Z \sim K_{{{\mathscr {X}}}/{{\mathscr {C}}}} + {{\mathscr {D}}}. \end{aligned}$$

In particular if $K_{{{\mathscr {X}}}/{{\mathscr {C}}}} + {{\mathscr {D}}}$ is big, then Z is a big divisor on ${{\mathscr {X}}}$.

Proof

By assumption $({{\mathscr {X}}},{{\mathscr {D}}})$ is log canonical, and therefore ${{\mathscr {X}}}$ is normal. Since we are assuming that ${{\mathscr {D}}}$ is a Cartier divisor, this implies that $K_{{\mathscr {X}}}$ is ${{\mathbb {Q}}}$-Cartier. Recall that, for the fibration $\rho :{{\mathscr {X}}}\rightarrow {{\mathscr {C}}}$, the canonical divisor of ${{\mathscr {X}}}$ verifies

$$\begin{aligned} K_{{\mathscr {X}}}= K_{{{\mathscr {X}}}/{{\mathscr {C}}}} + \rho ^* (K_{{\mathscr {C}}}). \end{aligned}$$

At the same time, the canonical divisor of is , and therefore the Riemann–Hurwitz formula for implies that

where ${\mathrm{Ram}}$ is the ramification divisor of $\pi$ and we used that . Therefore we obtain that

$$\begin{aligned} K_{{{\mathscr {X}}}/{{\mathscr {C}}}} - \pi ^*{\mathrm{pr}}_1^* (K_{{{\mathbb {P}}}^2}) = {\mathrm{Ram}}. \end{aligned}$$

Since , the pull-back $\pi ^*{\mathrm{pr}}_1^*(K_{{{\mathbb {P}}}^2})$ is linearly equivalent to $-({{\mathscr {D}}}+ {\mathrm{Ram}}_{{\mathscr {D}}})$, where ${\mathrm{Ram}}_{{\mathscr {D}}}$ is the ramification coming from the support of ${{\mathscr {D}}}$. Similarly, ${\mathrm{Ram}} = Z + {\mathrm{Ram}}_{{\mathscr {D}}}$ and therefore $K_{{{\mathscr {X}}}/{{\mathscr {C}}}} + {{\mathscr {D}}}\sim Z$, as wanted. $\square$

Lemma 6.1 was used in the split case to show that, if X is of log general type, then Z is a big divisor. However, in our situation, Lemma 6.1 only implies that the restriction of Z to the generic fiber is big, which is sometimes referred as Z is relatively big or $\rho$-big. This follows from the fact that

$$\begin{aligned} (K_{{{\mathscr {X}}}/{{\mathscr {C}}}} + {{\mathscr {D}}})\! \restriction _{{{\mathscr {X}}}_\eta } \cong K_{{{\mathscr {X}}}_\eta } + {{\mathscr {D}}}_\eta , \end{aligned}$$

and we are assuming that the generic fiber has the property that the log canonical divisor is big. This in general does not imply that Z is a big divisor (consider for example a trivial family over the base). Note however that in certain cases Z is already positive, as shown in the following remark.

Remark 6.2

One important example of applications of the main theorem is for complements of divisors in ${{\mathbb {P}}}^2$. In this case, and D is an effective member of the linear system . The condition that the fibers of $\rho = {\mathrm{pr}}_2$ are of log general type implies that $m \geqslant 4$ and the fact that the fibration is not isotrivial implies that $\deg _{{\mathscr {C}}}{{\mathscr {L}}}> 0$. Then a direct computation shows that and therefore

which is ample since $m \geqslant 4$ and $\deg _{{\mathscr {C}}}{{\mathscr {L}}}>0$. This shows that $Z = K_{{{\mathscr {X}}}/{{\mathscr {C}}}} + D$ is big in this case.

To obtain positivity properties of the divisor Z it is sufficient to add to it the pull-back of a divisor from the base curve. More specifically, it is enough to add to Z the pull-back of the divisor $K_{{\mathscr {C}}}+ 2 S$ on the curve ${{\mathscr {C}}}$. We note that, by hypothesis, $\#\,S \geqslant 2$ and therefore the divisor $K_{{\mathscr {C}}}+ 2S$ is very ample on ${{\mathscr {C}}}$.

Proposition 6.3

The divisor $Z + \rho ^* (K_{{\mathscr {C}}}+ 2S)$ is big. Moreover there exist $m>0$ that depends only on the generic fibers of $({{\mathscr {X}}},{{\mathscr {D}}})$ and Z such that $m(Z + \rho ^*(K_{{\mathscr {C}}}+ 2S)) \sim H + E$ with H a very ample divisor and E an effective divisor on ${{\mathscr {X}}}$.

Proof

The argument is somehow standard; we reproduce it here for completeness. Since Z is $\rho$-big, there exists a $\rho$-ample Cartier divisor $H_1$ and an effective Cartier divisor E on ${{\mathscr {X}}}$ such that $\rho _*{{\mathscr {O}}}_{{\mathscr {X}}}(E) \ne 0$ and $mZ \sim H_1 + E$ for some $m \geqslant 1$ that depends on X and the generic fiber of Z (see [23, Proposition 1.6.33]). Moreover, by [36, Lemma 01VU] we can assume that $H_1$ is in fact very ample up to replacing m by a large enough multiple. By our assumption on S we see that $K_{{\mathscr {C}}}+2S$ is very ample on ${{\mathscr {C}}}$, since its degree satisfies $\deg K_{{\mathscr {C}}}+ 2S \geqslant 2g + 1$. Then, by [23, Proposition 1.6.15(i)], the divisor $H_1 + \rho ^*(K_{{\mathscr {C}}}+ 2S)$ is very ample on ${{\mathscr {X}}}$, hence

$$\begin{aligned} m(Z + \rho ^*(K_{{\mathscr {C}}}+ 2S)) \sim H_1 + \rho ^*( m(K_{{\mathscr {C}}}+ 2S) ) + E. \end{aligned}$$

This shows that $Z + \rho ^*(K_{{\mathscr {C}}}+ 2S)$ is big and the desired equivalence holds with $H = H_1 + \rho ^* m(K_{{\mathscr {C}}}+ 2S)$. $\square$

Proposition 6.3 will be used in the next section to bound the degree of a section by estimating the intersection with the divisor Z. We end this section with a result that allows a more direct control on the positivity of Z when the family has nice properties coming from the theory of stable pairs.

When the log canonical divisor of the generic fiber of $\rho$ is ample, we can choose a model of (X, D) over ${{\mathscr {C}}}$ that is a stable family in the sense of Kollár (see for example [3, Definition 2.7]). In this situation we show that, when ${{\mathscr {D}}}$ is an ample divisor on ${{\mathscr {X}}}$, the divisor $K_{{{\mathscr {X}}}/{{\mathscr {C}}}} + {{\mathscr {D}}}$, and therefore Z, is indeed big. This follows from a result of Kovács and Patakfalvi in [27].

Proposition 6.4

Let $(X,D) \rightarrow B$ be a stable family such that the generic fiber has log canonical singularities, the base B is a nonsingular projective curve and D is an ample divisor on X. Then, $K_{X/B} + D$ is big.

Proof

Let E be an irreducible divisor in the linear system nD and consider the pair $\bigl (X,\frac{1}{n}E \bigr ) \rightarrow B$: it is a stable family where the generic fiber has now klt singularities (see [26, Definition 2.8]). The fact that D is ample and the base is one dimensional implies that the variation of the family is maximal and therefore [27, Corollary 7.3] implies that $K_{X/Y} + \frac{1}{n}E$ is big, finishing the proof. $\square$

Proposition 6.4 has the advantage of reducing the problem of testing whether the divisor Z is big to an ampleness condition on the model of the divisor ${{\mathscr {D}}}$, a condition that is easier to check in the applications.

7 Geometric Lang–Vojta for ramified covers of ${{\mathbb {G}}}_m^2$

In this section we apply Proposition 5.1 together with Proposition 6.3 to deduce Theorem A. This extends explicitly [20, Theorem 2] to the case of non-isotrivial pairs.

Theorem B

Let $\rho :({{\mathscr {X}}},{{\mathscr {D}}}) \rightarrow {{\mathscr {C}}}$ be a fibered threefold, let Z be the closure of the ramification divisor of the finite map as defined in Sect. 2.3, and let $Z_\eta$ be its generic fiber. Then, there exists a constant $C = C(Z_\eta , \deg \pi )$ such that, for every section $\sigma :{{\mathscr {C}}}\rightarrow {{\mathscr {X}}}$ with , the following holds:

Proof

By Proposition 6.3, the divisor $Z+\rho ^*(K_{{\mathscr {C}}}+ 2S)$ is a big ${{\mathbb {Q}}}$-Cartier divisor and there exist an integer m independent of ${{\mathscr {C}}}$, an effective divisor E (independent of ${{\mathscr {C}}}$) and a very ample divisor H on ${{\mathscr {X}}}$ such that $m(Z + \rho ^*(K_{{\mathscr {C}}}+ 2S)) \sim H + E$. We can estimate the degree of $\sigma ({{\mathscr {C}}})$ by estimating . Recall that the constant C implicitly depends on the projective embedding ${{\mathscr {X}}}\rightarrow {{\mathbb {P}}}^n$, and therefore on H.

First we notice that, if $\sigma ({{\mathscr {C}}})$ is contained in the support of E, then we obtain a bound of the desired form immediately. Indeed, assuming E is irreducible by arguing on every irreducible component, since $\sigma ({{\mathscr {C}}}) \subset E$, the restriction $\rho \vert _E$ is a flat dominant map where now E has relative dimension 1, i.e., E is a curve over the function field $\kappa ({{\mathscr {C}}})$. The generic fiber of the image $\pi (E)$ intersects $\pi (D)$ in at least three points, since our assumptions imply that the generic fiber of $\pi (Z)$ is disjoint from the singular points of . This shows that $E/\kappa ({{\mathscr {C}}})$ is a curve of log general type hence Siegel’s Theorem applies (e.g. in the form of [28]): in particular, either there are only finitely many integral sections $\sigma \in E(\kappa ({{\mathscr {C}}}))$ or E is isotrivial. In both cases we obtain directly that is bounded by $c_1 = c_1(Z_\eta , \deg \pi )$.

We are therefore left with the case in which $\sigma ({{\mathscr {C}}})$ is not contained in the support of E. Then,

(7.1)

Since $\sigma$ is a section we get that is equal to the degree of $(K_{{\mathscr {C}}}+ 2S)$, hence

Therefore it is enough to bound the intersection .

We view ${{\mathbb {G}}}_m^2$ embedded in ${{\mathbb {P}}}^2$ as the complement of the divisor $UVW=0$, where U, V and W are the homogeneous coordinates in ${{\mathbb {P}}}^2$. Since $\pi (Z_{\eta })$ avoids the singular points of , if we consider the morphism restricted to the complement of S (which by abuse of notation we still denote by ), there exist two S-units u, v such that is given by $( (u\,{:}\,v\,{:}\,1), {{\,\mathrm{id}\,}}_{{\mathscr {C}}})$.

Let $f(U,V,W) = 0$ be an equation for $\pi (Z)$, which is a polynomial with coefficients in $\kappa ({{\mathscr {C}}})$. Then, we can estimate the intersection by counting the number of zeros of f evaluated at (u, v, 1). Therefore (7.1) can be written as

(7.2)

On the other hand, if at least one of the heights of u and v is larger than the constant appearing in Proposition 5.1, we have that for every $\varepsilon > 0$ the following equality holds:

(7.3)

Up to a finite extension of S, independent of u and v, we can assume that the coefficients of f are S-units, and therefore f(u, v, 1) is an S-integer. Given the hypothesis on the intersection between $\pi (Z_{\eta })$ and the boundary , the constant term $f(0,0,1) = \mu$ is a nonzero S-unit (independent of u and v). We write f(u, v, 1) as sum of monomials using the following notation:

We get $w_1 + \cdots + w_n = 0$, where $w_n = \mu$. If a proper subsum vanishes, this gives an equation of the form $g(u,v) = 0$ which implies that $\deg \pi (\sigma ({\mathscr {C}}))$ is bounded by $\deg g \leqslant \deg f$ thus finishing the proof. In particular, this might happen when u and v are multiplicatively dependent, and one obtains a bound on the exponents of the multiplicative relation.

Therefore, we can assume that no proper subsum of $w_1 + \cdots + w_n = 0$ vanishes, so that we can apply [7, Theorem B] to obtain

(7.4)

where

$$\begin{aligned} \gamma _0 = 0, \qquad \gamma _{\ell } = \frac{1}{2}\, (\ell -1)(\ell -2) \quad \text {for all}\;\; \ell \geqslant 1, \end{aligned}$$

and $m_{P}$ denotes the number of $w_i$s that are units at P. The right-hand side of (7.4) can be bounded from above obtaining

(7.5)

Notice that $\gamma _{m_P} = \gamma _n$ for all but finitely many . In this finite set, which is the set of zeros of f(u, v, 1) outside S, we have $\gamma _{m_P} = \gamma _{n-1}$. Therefore we can rewrite equation (7.5) as

Using (7.3) we have that, for every $\varepsilon > 0$,

(7.6)

Since $w_1=-f(u,v,1)$ and $w_n=\mu$, we can bound from below the projective height as

(7.7)

Combining (7.2) with (7.7), we have

(7.8)

Using (7.6) with $\varepsilon = \dfrac{1}{ 2m(n-2) \deg \pi }$ together with (7.8) we obtain

(7.9)

where

and $c_3$ is the constant appearing in Proposition 5.1, which depends only on $Z_\eta$ and on $\deg \pi$.

Finally, we obtained that either the degree of $\sigma ({\mathscr {C}})$ is directly bounded by a constant which depends only on $Z_\eta$ and on the degree of $\pi$, or the maximum of the heights of u and v is bounded by (7.9), which implies a bound for the degree of the image $\sigma ({{\mathscr {C}}})$. $\square$

8 Applications

In this section we give an explicit application of Theorem A, which generalizes [20, Theorem 1] and [37, Theorem 1.3].

Theorem B Let ${{\mathscr {D}}}$ be a non-isotrivial stable flat family of divisors in ${{\mathbb {P}}}^2$ over a smooth integral curve ${{\mathscr {C}}}$. Assume that there exists a finite set of points $S \subset {{\mathscr {C}}}$ such that, for every , the fiber ${{\mathscr {D}}}_P$ of ${{\mathscr {D}}}$ over P has simple normal crossings singularities and $r \geqslant 3$ components of total degree $d \geqslant 4$. Then, there exists a constant C, depending only on the generic fiber of ${{\mathscr {D}}}$, such that for every section verifying the following holds:

Proof

By Remark 6.2, ${{\mathscr {D}}}$ is an ample (Cartier) divisor on , and therefore we can assume that for an ample and $d \geqslant 4$. Let $0 \in {{\mathscr {C}}}$ be the generic point and let $D^0$ be the generic fiber of the family ${{\mathscr {D}}}\rightarrow {{\mathscr {C}}}$. The hypotheses ensure that we can write $D^0 = D_1 + D_2 + \dots + D_r$, with $r \geqslant 3$. Let $g_{1}$ be an equation of $D_1$, $g_{2}$ be an equation of $D_2$, and $g_{3}$ be an equation of $D_3 + \dots + D_r$, of degree respectively $d_1, d_2$ and $d_3$; clearly $d_1 + d_2 + d_3 = d$. Let us consider the map

$$\begin{aligned}&\pi _0:{{\mathbb {P}}}^2 \rightarrow {{\mathbb {P}}}^2 \\&\pi _0([x_0\,{:}\,x_1\,{:}\,x_2]) = \bigl [g_1^{d_2d_3}([x_0\,{:}\,x_1\,{:}\,x_2])\,{:}\, g_2^{d_1d_3}([x_0\,{:}\,x_1\,{:}\,x_2])\,{:}\, g_3^{d_1d_2}([x_0\,{:}\,x_1\,{:}\,x_2])\bigr ], \end{aligned}$$

which is clearly a finite map and it is defined everywhere since $D^0$ has simple normal crossings singularities. Moreover and therefore $\pi _0$ restricts to a finite cover of ${{\mathbb {G}}}_m^2$ on . The same argument as in [20, Theorem 1] implies that if E is an irreducible component of the ramification divisor of $\pi _0$ such that $\pi _0(E)$ passes through a singular point of then E lies in the support of $D^0$. Therefore, we can extend $\pi _0$ to a generically finite map which, after possibly resolving indeterminacy of the map, fits into the diagram

By the above discussion, the ramification divisor of $\pi$ has the property that every irreducible component E such that ${\mathrm{pr}}_1(\pi (E))$ passes through one of the singular points of is contained in the support of ${{\mathscr {D}}}$; in particular, is contained in S, where Z is the ramification divisor of . Therefore, we can apply Theorem A whose conclusion implies the theorem. $\square$

We finally give an explicit example where Theorem B applies.

Example 8.1

Consider the one-parameter family of reducible plane quartics given by the vanishing of

$$\begin{aligned} (y_0x_0)(y_0x_1)\bigl (y_0^2x_2^2-y_0^2x_1^2-y_1^2x_1x_0-y_0^2x_0^2\bigr ), \end{aligned}$$

in , and let be the associated divisor. Let S be the finite set of points of ${{\mathbb {P}}}^1$ for which the specialized divisor has no normal crossings singularities or does not have three irreducible components; note that S contains at least two points, e.g. $[0\,{:}\,1]$ and $[1\,{:}\,\sqrt{2}]$. We define as in Theorem B

$$\begin{aligned} \pi _0([x_0 \,{:}\, x_1 \,{:}\, x_2]) = \bigl [(y_0x_0)^2 \,{:}\, (y_0x_1)^2 \,{:}\, y_0^2x_2^2 - y_0^2x_1^2 - y_1^2x_1 x_0 - y_0^2x_0^2\bigr ], \end{aligned}$$

which is clearly a finite map and well defined outside of the generic fiber $D^0$ of ${{\mathscr {D}}}$. In this case, one can compute explicitly the ramification of the finite map $\pi _0$ which is given by $(y_0x_0)(y_0x_1)(y_0^2x_2) = 0$; in particular, it has three irreducible components. Two of them, namely $y_0x_0 = 0$ and $y_0x_1 = 0$, are contained in the support of $D^0$, while $y_0^2x_2 = 0$ corresponds to the generic fiber of Z in our previous notation. We see directly that Z is an ample divisor lying in the linear system and moreover $Z \sim K_{{{\mathbb {P}}}^2 \times {{\mathbb {P}}}^1} + {{\mathscr {D}}}$. Clearly, since S contains the point $[0\,{:}\,1]$, one sees directly that is contained in S, and therefore we can apply Theorem B obtaining a bound on the degree of the image of sections such that is contained in S.