1 Introduction
There are several bounds for greatest common divisors in a variety of contexts. Bugeaud et al. [Reference Bugeaud, Corvaja and Zannier2] gave an upper bound for the greatest common divisor of
$a^n-1$
and
$b^n-1$
, where
$a,b$
are multiplicatively independent integers. Corvaja and Zannier [Reference Corvaja and Zannier3] generalised this result when
$a^n$
and
$b^n$
are replaced by elements of a fixed finitely generated subgroup of
$\overline {\mathbb {Q}}^\ast $
. Levin [Reference Levin10] gave a greatest common divisor bound for polynomials in several variables with algebraic coefficients, generalising the result obtained by Corvaja and Zannier.
The greatest common divisor (GCD) problem has been studied in various other settings. A similar result to that by Bugeaud, Corvaja and Zannier was stated and proved by Ailon and Rudnick [Reference Ailon and Rudnick1] for nonconstant and multiplicatively independent complex polynomials. Some extensions of this last work were given by Ostafe in [Reference Ostafe12]. In addition, a positive characteristic version was established by Ghioca et al. in [Reference Ghioca, Hsia and Tucker6].
In the context of Nevanlinna theory, Pastén and Wang [Reference Pasten and Wang13] gave a GCD bound for algebraically independent meromorphic functions over
${\mathbb {C}}$
. A stronger result of this type was obtained by Levin and Wang (see [Reference Levin and Wang11, Theorem 1.3]).
We study the GCD problem in the context of varieties over number fields. Let K be a number field, X a smooth projective variety defined over K and
$\mathcal {L}$
a line sheaf on X. There is a height function
$h_{X,\mathcal {L}}:X(\overline {K})\to \mathbb {R}$
(see for instance [Reference Vojta15]). For a divisor D on X, we simply write
$h_{X,D}$
instead of
$h_{X,\mathcal {O}(D)}$
.
This height in the case of blow-ups is closely related to the generalised greatest common divisor (for the details, see [Reference Silverman14]).
Definition 1.1 [Reference Silverman14, Definition 2].
Let X be a smooth variety defined over K and
$Y\subset X$
be an irreducible subvariety of X also defined over K, of co-dimension
$c\geq 2$
. Let
$\pi :\tilde {X}\to X$
be the blow up of X along Y and let
$E_Y$
be the exceptional divisor of the blow up. For
$x\in X\setminus Y$
, let
$\tilde {x}=\pi ^{-1}(x)$
. The generalised (logarithmic) greatest common divisor of the point
$x\in (X\setminus Y)(\overline {K})$
with respect to Y is
$$ \begin{align*}h_{\gcd}(x;Y)=h_{\tilde{X},E_Y}(\tilde{x}).\end{align*} $$
For some examples of this quantity, see [Reference Silverman14]. Silvermann [Reference Silverman14] noticed that the generalised GCD is closely related to Vojta’s main conjecture. Bounds for the generalised GCD in varieties were given by Grieve [Reference Grieve7] for rational points in certain toric varieties and by Wang and Yasukufu [Reference Wang and Yasufuku16] under integrality conditions on the points. Recently, García-Fritz and Pastén [Reference García-Fritz and Pastén5] gave an upper bound when the closed sub-scheme is reduced and consists on d geometric points, without integrality conditions.
Theorem 1.2 [Reference García-Fritz and Pastén5, Theorem 3.1].
Let X be a smooth projective variety defined over a number field K of dimension n and let
$\mathcal {A}$
be an ample line sheaf on X. Let Y be a reduced closed sub-scheme consisting of d geometric points. Then, given any
$\varepsilon>0$
, there is a properly contained Zariski closed set
$Z_\varepsilon \subset X$
such that
$$ \begin{align*}h_{\gcd}(x;Y)\leq \Bigg(\sqrt[n]{\frac{d}{(\mathcal{A}^n)}}+\varepsilon\Bigg)h(\mathcal{A},x)+O(1)\end{align*} $$
as x varies in
$(X-Z_{\varepsilon })(\overline {K})$
.
The goal of this work is to give an upper bound for
$h_{\gcd }(x;Y)$
. We follow the ideas of García-Fritz and Pastén to give an upper bound for the GCD when the closed sub-scheme is a higher dimensional irreducible sub-variety. Our main result is as follows.
Theorem 1.3 (GCD bound).
Let X be a smooth projective variety defined over a number field K of dimension n and let Y be an irreducible sub-variety of dimension d, also defined over K. Let
$X'=X\setminus Y$
and
$c=n-d$
. Then, given any ample line sheaf
$\mathcal {A}$
and
$\varepsilon>0$
, there is a properly contained Zariski closed set
$Z_\varepsilon \subset X'$
such that, for all
$x\in (X'\setminus Z_\varepsilon )(\overline {K})$
,
$$ \begin{align*}h_{\gcd}(x;Y)\leq \bigg(\bigg(\frac{(\mathcal{A}^d\cdot Y)}{(\mathcal{A}^n)}\cdot\frac{n!}{c!}\bigg)^{{1}/{c}}+\varepsilon\bigg)h(\mathcal{A},x)+O_{\varepsilon}(1).\end{align*} $$
The zero-dimensional case studied by García-Fritz and Pastén has interesting applications in terms of the Bombieri–Lang conjecture and Vojta’s conjecture (see [Reference García-Fritz and Pastén5, Theorems 1.4 and 3.2]). We expect similar consequences from Theorem 1.3, which we leave for future research.
The proof of Theorem 1.3 is inspired by the methods of [Reference García-Fritz and Pastén5]. However, new complications appear and we need to prove an auxiliary result on the asymptotic Riemann–Roch theorem (see Theorem 2.3).
2 Riemann–Roch
In this section, we will give an estimate of the dimension of a certain space that will be useful in proving the GCD bound. We recall the asymptotic Riemann–Roch theorem.
Theorem 2.1 (Asymptotic Riemann–Roch).
Let X be a projective variety of dimension n,
$\mathcal {L}$
an ample line sheaf on X and
$\mathcal {F}$
a coherent sheaf on X. Then,
$$ \begin{align*} h^i(X,\mathcal{L}^{\otimes m}\otimes \mathcal{F})&=O(m^{n-1}) \quad\text{for}~i>0, \\ h^0(X,\mathcal{L}^{\otimes m}\otimes \mathcal{F})&=\frac{(\mathcal{L}^n\cdot \mathcal{F})}{n!}m^n+O(m^{n-1}), \end{align*} $$
where the constant in
$O(m^{n-1})$
depends on
$X,\mathcal {F}$
and
$\mathcal {L}$
.
Suppose that X is a projective variety over a field K and
$Y\subset X$
is a subvariety of X also defined over K. Then, we have the ideal sheaf
$\mathcal {I}_Y$
associated to Y and, for
$r\geq 1$
, we consider the coherent sheaf
$\mathcal {O}_X/\mathcal {I}_Y^r$
. In Section 3, we will need to estimate the quantity
$h^0(X,\mathcal {L}^{\otimes m}\otimes \mathcal {O}_X/\mathcal {I}_Y^r)$
and we will have to know how this dimension changes as r increases. This information is not provided by Theorem 2.1. Nevertheless, the following two results will help us to proceed.
Lemma 2.2. Let Z be a projective scheme of finite type over a number field K of dimension d. Let
$\mathcal {A}$
be an ample base point free line sheaf on Z. Then,
$$ \begin{align*}h^0(Z,\mathcal{A})\leq (\mathcal{A}^d)+d.\end{align*} $$
Proof. Let
$q:Z\to \mathbb {P}^{n}$
be the morphism defined by
$\mathcal {A}$
and let
$Y=q(Z)$
be its image. Then,
$h^0(Z,\mathcal {A})=n+1$
. Let
$n_Y$
be the dimension of Y.
First, we will prove that
$n+1\leq (\mathcal {O}_{\mathbb {P}^n}(1)^{n_Y}\cdot Y)+n_Y$
. We will do this by induction on
$n_Y$
. If
$n_Y=0$
, then the inequality holds. If
$n_Y\geq 1$
, let
$H\subset \mathbb {P}^n$
be a hyperplane meeting Y properly. Consider
$Y'=Y\cap H\subset \mathbb {P}^{n-1}$
. Then, there is a properly contained
$Z'\subset Z$
and a morphism
$Z'\to Y'\subset \mathbb {P}^n$
. By [Reference Hartshorne8, Theorem 7.1, II], this morphism comes from an ample and base point free line sheaf
$\mathcal {A}'$
on
$Z'$
. Then, by induction,
$$ \begin{align*}(n-1)+1\leq (\mathcal{O}_{\mathbb{P}^{n-1}}(1)^{n_Y-1}\cdot Y')+(n_Y-1) \Rightarrow n+1 \leq (\mathcal{O}_{\mathbb{P}^{n-1}}(1)^{n_Y-1}\cdot Y')+n_Y.\end{align*} $$
Furthermore,
$(\mathcal {O}_{\mathbb {P}^{n-1}}(1)\cdot Y')=(\mathcal {O}_{\mathbb {P}^{n-1}}\cdot H\cap Y)=(\mathcal {O}_{\mathbb {P}^n}(1)\cdot Y)$
. Therefore,
$$ \begin{align*}n+1\leq (\mathcal{O}_{\mathbb{P}^n}(1)\cdot Y)+n_Y.\end{align*} $$
Finally, by [Reference Lazarsfeld9, Corollary 1.2.15], the morphism
$q:Z\to Y$
is finite. Thus,
$q_{\ast }Z=\deg (q)\cdot Y$
in
$K_{d}(\mathbb {P}^n)$
. By the projection formula,
$(A^d\cdot Z)=t(\mathcal {O}_{\mathbb {P}^n}(1)\cdot Y)$
. Since
$n_y\leq d$
, the desired conclusion follows.
Theorem 2.3 (Riemann–Roch-type inequality).
Let X be a projective variety defined over K and
$Y\subset X$
be an irreducible subvariety of dimension d. Let
$\mathcal {I}_Y$
be the ideal sheaf associated to Y. Then, given any very ample line sheaf
$\mathcal {A}$
on X,
$$ \begin{align*}h^0(X,\mathcal{A}^{\otimes m}\otimes \mathcal{O}_X/\mathcal{I}_Y^r)\leq \frac{r^c\cdot m^{d}}{c!}e_X(Y)\cdot(\mathcal{A}^{d}\cdot Y)+O(r^{c-1})\end{align*} $$
for
$r\gg 1$
, where
$c=\dim X-d$
,
$e_{Y}(X)$
is the algebraic multiplicity of X along Y, and the constant in
$O(r^{c-1})$
only depends on
$X,Y$
and
$\mathcal {A}$
.
Proof. Since
$\mathcal {A}$
is very ample, then
$\mathcal {A}|_{Y_r}$
is very ample on
$Y_r$
. By Lemma 2.2,
$$ \begin{align} h^0(X,\mathcal{A}^{\otimes m}\otimes \mathcal{O}_X/\mathcal{I}_Y^r)\leq (\mathcal{A}^d\cdot \mathcal{O}_X/\mathcal{I}_Y^r)m^d+d. \end{align} $$
Let
$\xi _Y$
be the generic point of Y. Then,
$$ \begin{align*}(\mathcal{A}^d\cdot\mathcal{O}_X/\mathcal{I}_Y^r)=\mathrm{length}_{\xi_Y}(\mathcal{O}_X/\mathcal{I}_Y^r)_{\xi_Y}(\mathcal{A}^d\cdot Y).\end{align*} $$
By [Reference Fulton4, Example 4.3.4], this length is equal to
$$ \begin{align*}e_Y(X)\cdot\frac{r^c}{c!}+O(r^{c-1})\end{align*} $$
for
$r\gg 1$
. Substituting this in (2.1) gives the result.
3 Proof of the GCD bound
In this section, we obtain a bound for the generalised greatest common divisor. Throughout this section:
-
• K is a number field;
-
• X is a smooth projective variety defined over K;
-
• Y is an irreducible subvariety of X of codimension
$c\geq 2$
also defined over K; -
•
$X'=X\setminus Y$
.
Proposition 3.1. For an ample line sheaf
$\mathcal {A}$
on X, if there is an effective divisor D on X such that
$\mathcal {O}(D)=\mathcal {A}^{\otimes m}$
and
$r=m_Y(D)$
for some positive integers
$m,r$
, then there is a properly contained Zariski closed set
$Z\subset X$
such that for all
$x\in (X'\setminus Z)(\overline {K})$
,
$$ \begin{align*}h_{\gcd}(x;Z)\leq \frac{m}{r}h(\mathcal{A},x)+O(1).\end{align*} $$
Proof. Let
$E_Y$
be the exceptional divisor of the blow-up of X along Y and let B be the base locus of
$\tilde {D}$
. By the assumption on D, given any
$y\in (\tilde {X}\setminus B)(\overline {K})$
, we can find x such that
$\pi (x)=y$
. Then,
$$ \begin{align*} r\cdot h_{\gcd}(x;Y) &= r\cdot h_{\tilde{X},E_Y}(y)+O(1) \\ &\leq h_{\tilde{X},\pi^{\ast}(D)}(y)+O(1) \\ &= h_{X,D}(x)+O(1)\\ &= m\cdot h_{X,\mathcal{A}}(x)+O(1). \end{align*} $$
Thus,
$$ \begin{align*}h_{\gcd}(x;Y)\leq \frac{m}{r}h_{X,\mathcal{A}}(x)+O(1)\end{align*} $$
for all
$x\in (X'\setminus Z)(\overline {K})$
with
$Z=\pi (B)$
.
Theorem 3.2. Given any ample line sheaf
$\mathcal {A}$
on X and
$\varepsilon>0$
, there is a properly contained Zariski closed set
$Z_\varepsilon \subset X$
such that for all
$x\in (X'\setminus Z_\varepsilon )(\overline {K})$
,
$$ \begin{align*}h_{\gcd}(x;Y)\leq \bigg(\bigg(\frac{(\mathcal{A}^d\cdot Y)}{(\mathcal{A}^n)}\cdot\frac{n!}{c!}\bigg)^{{1}/{c}}+\varepsilon\bigg)h_{X,\mathcal{A}}(x)+O_{\varepsilon}(1).\end{align*} $$
Proof. First, consider the case when
$\mathcal {A}$
is very ample. By Proposition 3.1, we have to guarantee the existence of a divisor D in X such that
$\mathcal {O}(D)=\mathcal {A}^{\otimes m}$
and
$r=m_Y(D)$
for some positive integers
$m,r$
.
Let m and r be positive integers and
$\mathcal {I}_Y$
be the ideal sheaf associated to Y. Then, the associated exact sequence to
$\mathcal {I}_Y^r$
is
$$ \begin{align*} 0\to \mathcal{I}_Y^r\to \mathcal{O}_X\to \mathcal{O}_X/\mathcal{I}_Y^r\to 0. \end{align*} $$
Tensoring by
$\mathcal {A}^{\otimes m}$
and passing to global sections yields
$$ \begin{align*} 0\to H^0(X,\mathcal{A}^{\otimes m}\otimes \mathcal{I}_Y^r)\to H^0(X,\mathcal{A}^{\otimes m})\to H^0(X,\mathcal{A}^{\otimes m}\otimes \mathcal{O}_X/\mathcal{I}_Y^r). \end{align*} $$
If
$h^0(X,\mathcal {A}^{\otimes m})> h^0(X,\mathcal {A}^{\otimes m}\otimes \mathcal {O}_X/\mathcal {I}_Y^r)$
, then
$H^0(X,\mathcal {A}^{\otimes m}\otimes \mathcal {I}_Y^r)$
is nontrivial and we obtain the desired divisor (in fact, we obtain
$m_Y(D)\geq r$
but we do not need this).
Now, given any
$\varepsilon>0$
, there is a sufficiently large choice for
$m,r$
such that
$$ \begin{align*}\frac{(\mathcal{A}^n)}{n!}m^n> \frac{r^c\cdot m^d}{c!}(\mathcal{A}^d\cdot Y) \quad \mathrm{and} \quad \frac{m}{r}<\bigg(\frac{(\mathcal{A}^d\cdot Y)}{(\mathcal{A}^n)}\cdot\frac{n!}{c!}\bigg)^{{1}/{c}}+\varepsilon.\end{align*} $$
By Theorems 2.1 and 2.3, when
$\mathcal {A}$
is very ample, we obtain the result.
Finally, if
$\mathcal {A}$
is ample, consider an integer k such that
$\mathcal {A}^{\otimes k}$
is very ample. Applying the result to
$\mathcal {A}^{\otimes k}$
gives
$$ \begin{align*} h_{\gcd}(x;Y)&\leq \bigg(\bigg(\frac{((\mathcal{A}^{\otimes k})^d\cdot Y)}{((\mathcal{A}^{\otimes k})^n)}\cdot\frac{n!}{c!}\bigg)^{{1}/{c}}+\varepsilon\bigg)h_{X,\mathcal{A}^{\otimes k}}(x)+O_{\varepsilon}(1) \\ &\leq \bigg(\bigg(\frac{k^d(\mathcal{A}^d\cdot Y)}{k^n(\mathcal{A}^n)}\cdot\frac{n!}{c!}\bigg)^{{1}/{c}}+\varepsilon\bigg)k\cdot h_{X,\mathcal{A}}(x)+O_{\varepsilon}(1) \\ &\leq \bigg(\frac{1}{k}\bigg(\frac{(\mathcal{A}^d\cdot Y)}{(\mathcal{A}^n)}\cdot\frac{n!}{c!}\bigg)^{{1}/{c}}+\varepsilon\bigg)k\cdot h_{X,\mathcal{A}}(x)+O_{\varepsilon}(1), \end{align*} $$
which gives the desired result.
Acknowledgements
This work was done in the context of my Master’s thesis. Because of this, I would like to thank my advisor Héctor Pastén and the readers Siddarth Mathur and Ricardo Menares for their comments. I also thank the referee for suggestions to improve the content of this article.
