Anomalous subvarieties – structure theorems and applications.

*(English)*Zbl 1145.11049Let \(T=\mathbb{G}_{m}^{n}\) be an \(n\)-dimensional algebraic torus over \(\overline{\mathbb{Q}}\), where \(n\geq 2\). For \(0\leq r\leq n-1\), let \(\mathcal{H}_{\,r}\) denote the union of all \(r\)-dimensional algebraic subgroups of \(T\). Set \(\mathcal{H}=\bigcup_{\,r=0}^{\,n-1}\mathcal{H}_{\,r}\).

In [[1]: E. Bombieri, D. Masser and U. Zannier, “Intersecting a curve with algebraic subgroups of multiplicative groups”, Int. Math. Res. Not. 1999, No. 20, 1119–1140 (1999; Zbl 0938.11031)], the authors studied the intersection of a given closed and irreducible curve \(C\subset T\) with \(\mathcal{H}\). In particular, they showed that if \(C\) is not contained in a translate of a proper subtorus of \(T\), then the set \(C\cap\mathcal{H}(\overline{\mathbb{Q}})\) has bounded (Weil) height. The paper under review is concerned with a generalization of this result to higher-dimensional subvarieties \(X\subset T\). To explain the sort of results actually proved in this paper, we go back to the works [[2]: E. Bombieri and U. Zannier, “Algebraic points on subvarieties of \(\mathbb{G}_{m}^{n}\)”, Int. Math. Res. Not. 1995, No. 7, 333–347 (1995; Zbl 0848.11030)] and [[3]: U. Zannier, “Appendix”, in “Polynomials with special regard to reducibility. With an Appendix by Umberto Zannier” by A. Schinzel. Encyclopedia of Mathematics and its Applications 77. Cambridge University Press (2000; Zbl 0956.12001), pp. 517–539] by the first and third authors.

Let \(X^{\circ}\) denote the complement in \(X\) of the union of all subvarieties of \(X\) which are translates of nontrivial subtori of \(T\). Then \(X^{\circ}\) is a Zariski-open subset of \(X\) [2]. Further, if \(X\) is defined over \(\overline{\mathbb{Q}}\), then \(X^{\circ}\cap\mathcal H_{1}\) is a set of bounded height [3]. In the paper under review the authors introduce a new set \(X^{{\circ}a}\), analogous to \(X^{\circ}\) and contained in it if \(X\neq T\), and show that it is (Zariski) open in \(X\) (in fact, a sharper “structure theorem” for \(X^{{\circ}a}\) is obtained. See Theorem 1.4 of the paper). The set \(X^{{\circ}a}\) is defined as the complement in \(X\) of the union of all “anomalous” subvarieties of \(X\). A positive-dimensional irreducible subvariety \(Y\) of \(X\) is anomalous if it lies in a translate \(K\) of an algebraic subgroup of \(T\) and its dimension is strictly larger than \(\text{dim}\, X+\text{dim}\,K-n\) (so \(X\) and \(K\) do not meet properly since \(\text{dim}\,(X\cap K)\geq \text{dim}\, Y>\text{dim}\,X+\text{dim}\,K-n\)).

The authors also state the following Bounded Height Conjecture. Let \(X\) be an irreducible subvariety of \(T\) of dimension \(r\). Then \(X^{{\circ}a}\cap\mathcal H_{\,n-r}\) is a set of bounded height. When \(X\) is a curve \(C\) not contained in a translate of dimension \(n-1\), then \(X^{{\circ}a}=C\) and the conjecture is true by the result on curves quoted above. On the other hand, if \(r=n-1\) (i.e., if \(X\) is a hypersurface in \(T\)) then \(X^{{\circ}a}=X^{\circ}\) and the conjecture is true by the result from [3] cited above. No other instances where the conjecture is true are known, but the authors have promised to settle the case of planes in \(T\) in a subsequent publication.

The second main result of the paper establishes the existence of a finite collection \(\Psi\) of translations \(S\) of tori by torsion points, satisfying \(\text{dim}(X\cap S)\geq \text{dim}\,S-1\), such that \(X\cap\mathcal H_{\,1}=\bigcup_{S\in\Psi}(X\cap S)\cap \mathcal H_{\,1}\). The significance of this result is that it reduces the problem of describing \(X\cap\mathcal H_{\,1}\) for general \(X\) to the hypesurface case since \(X\cap S\) may be regarded as a hypersurface in \(S\) and \(S\) is essentially \(\mathbb G_{m}^{d}\) for some \(d\). No analogous description of \(X\cap\,\mathcal H_{\,2}\) is known at present. The paper also contains a result (Theorem 1.6) on lacunary polynomials with algebraic coefficients which has implications for irreducibility questions. This result (which, in the interest of brevity, we do not state here) extends work of Schinzel and of the first and third authors in [3].

The paper also discusses a set \(X^{ta}\) which is obtained by removing from \(X\) all “torsion-anomalous” subvarieties of \(X\) (to define a torsion-anomalous subvariety, simply repeat the definition of “anomalous” above specializing \(K\) to a translate of the form \(gH\), where \(g\) is a torsion element of \(T\) and \(H\) is an algebraic subgroup of \(T\)). The following conjectures are discussed: (a) Let \(X\) be an irreducible subvariety of \(T\) defined over \(\mathbb C\). Then \(X^{ta}\) is Zariski-open in \(X\), and (b) if \(X\) (as in (a)) has dimension \(r\), then \(X^{ta}\cap\mathcal H_{n-r-1}\) is a finite set. The authors also discuss generalizations of the above conjectures to the case of semi-abelian varieties. Finally, the third author corrects an inaccuracy which appears in the proof of Theorem 2 in [3].

In [[1]: E. Bombieri, D. Masser and U. Zannier, “Intersecting a curve with algebraic subgroups of multiplicative groups”, Int. Math. Res. Not. 1999, No. 20, 1119–1140 (1999; Zbl 0938.11031)], the authors studied the intersection of a given closed and irreducible curve \(C\subset T\) with \(\mathcal{H}\). In particular, they showed that if \(C\) is not contained in a translate of a proper subtorus of \(T\), then the set \(C\cap\mathcal{H}(\overline{\mathbb{Q}})\) has bounded (Weil) height. The paper under review is concerned with a generalization of this result to higher-dimensional subvarieties \(X\subset T\). To explain the sort of results actually proved in this paper, we go back to the works [[2]: E. Bombieri and U. Zannier, “Algebraic points on subvarieties of \(\mathbb{G}_{m}^{n}\)”, Int. Math. Res. Not. 1995, No. 7, 333–347 (1995; Zbl 0848.11030)] and [[3]: U. Zannier, “Appendix”, in “Polynomials with special regard to reducibility. With an Appendix by Umberto Zannier” by A. Schinzel. Encyclopedia of Mathematics and its Applications 77. Cambridge University Press (2000; Zbl 0956.12001), pp. 517–539] by the first and third authors.

Let \(X^{\circ}\) denote the complement in \(X\) of the union of all subvarieties of \(X\) which are translates of nontrivial subtori of \(T\). Then \(X^{\circ}\) is a Zariski-open subset of \(X\) [2]. Further, if \(X\) is defined over \(\overline{\mathbb{Q}}\), then \(X^{\circ}\cap\mathcal H_{1}\) is a set of bounded height [3]. In the paper under review the authors introduce a new set \(X^{{\circ}a}\), analogous to \(X^{\circ}\) and contained in it if \(X\neq T\), and show that it is (Zariski) open in \(X\) (in fact, a sharper “structure theorem” for \(X^{{\circ}a}\) is obtained. See Theorem 1.4 of the paper). The set \(X^{{\circ}a}\) is defined as the complement in \(X\) of the union of all “anomalous” subvarieties of \(X\). A positive-dimensional irreducible subvariety \(Y\) of \(X\) is anomalous if it lies in a translate \(K\) of an algebraic subgroup of \(T\) and its dimension is strictly larger than \(\text{dim}\, X+\text{dim}\,K-n\) (so \(X\) and \(K\) do not meet properly since \(\text{dim}\,(X\cap K)\geq \text{dim}\, Y>\text{dim}\,X+\text{dim}\,K-n\)).

The authors also state the following Bounded Height Conjecture. Let \(X\) be an irreducible subvariety of \(T\) of dimension \(r\). Then \(X^{{\circ}a}\cap\mathcal H_{\,n-r}\) is a set of bounded height. When \(X\) is a curve \(C\) not contained in a translate of dimension \(n-1\), then \(X^{{\circ}a}=C\) and the conjecture is true by the result on curves quoted above. On the other hand, if \(r=n-1\) (i.e., if \(X\) is a hypersurface in \(T\)) then \(X^{{\circ}a}=X^{\circ}\) and the conjecture is true by the result from [3] cited above. No other instances where the conjecture is true are known, but the authors have promised to settle the case of planes in \(T\) in a subsequent publication.

The second main result of the paper establishes the existence of a finite collection \(\Psi\) of translations \(S\) of tori by torsion points, satisfying \(\text{dim}(X\cap S)\geq \text{dim}\,S-1\), such that \(X\cap\mathcal H_{\,1}=\bigcup_{S\in\Psi}(X\cap S)\cap \mathcal H_{\,1}\). The significance of this result is that it reduces the problem of describing \(X\cap\mathcal H_{\,1}\) for general \(X\) to the hypesurface case since \(X\cap S\) may be regarded as a hypersurface in \(S\) and \(S\) is essentially \(\mathbb G_{m}^{d}\) for some \(d\). No analogous description of \(X\cap\,\mathcal H_{\,2}\) is known at present. The paper also contains a result (Theorem 1.6) on lacunary polynomials with algebraic coefficients which has implications for irreducibility questions. This result (which, in the interest of brevity, we do not state here) extends work of Schinzel and of the first and third authors in [3].

The paper also discusses a set \(X^{ta}\) which is obtained by removing from \(X\) all “torsion-anomalous” subvarieties of \(X\) (to define a torsion-anomalous subvariety, simply repeat the definition of “anomalous” above specializing \(K\) to a translate of the form \(gH\), where \(g\) is a torsion element of \(T\) and \(H\) is an algebraic subgroup of \(T\)). The following conjectures are discussed: (a) Let \(X\) be an irreducible subvariety of \(T\) defined over \(\mathbb C\). Then \(X^{ta}\) is Zariski-open in \(X\), and (b) if \(X\) (as in (a)) has dimension \(r\), then \(X^{ta}\cap\mathcal H_{n-r-1}\) is a finite set. The authors also discuss generalizations of the above conjectures to the case of semi-abelian varieties. Finally, the third author corrects an inaccuracy which appears in the proof of Theorem 2 in [3].

Reviewer: Cristian D. Gonzales-Aviles (La Serena)