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Seshadri constants, diophantine approximation, and Roth’s theorem for arbitrary varieties

Abstract

In this paper, we associate an invariant \(\alpha _{x}(L)\) to an algebraic point \(x\) on an algebraic variety \(X\) with an ample line bundle \(L\). The invariant \(\alpha \) measures how well \(x\) can be approximated by rational points on \(X\), with respect to the height function associated to \(L\). We show that this invariant is closely related to the Seshadri constant \(\epsilon _{x}(L)\) measuring local positivity of \(L\) at \(x\), and in particular that Roth’s theorem on \(\mathbb {P}^1\) generalizes as an inequality between these two invariants valid for arbitrary projective varieties.

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Notes

  1. All uses of “Roth” as an adjective in this paper are in homage to the theorem proved by Klaus F. Roth and its later extensions by Ridout and Lang, and do not refer to the second named author of the paper.

  2. The extra quantifier “\(C\)” in Definition 2.10 can be absorbed by the condition that the finiteness is supposed to hold for all \(\gamma <\alpha _x\). The purpose of this quantifier in Definition 2.10 is to simplify arguments.

  3. To the best of our knowledge, the number \(\beta _x(L)\) was first defined by Per Salberger in unpublished work dating from 2006, where it was used to improve results of R. Heath-Brown on uniform upper bounds for the number of rational points of bounded height. Salberger also proved Corollary 4.2 as a key step in this work.

  4. As well as Propositions 2.14 (f) and 3.4 (c), and Corollary 4.4...

  5. If \(X\) is not normal, \(\Gamma (X,mL)\otimes _{k}K^{(v)}\) may only be a proper subspace of \(\Gamma (\widetilde{X}^{(v)},mL^{(v)}_{0})\). However, since the volume is a birational invariant, the asymptotic calculations go through without change and we omit further mention of this detail.

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Acknowledgments

We thank Chris Dionne, Laurence Ein, Robert Lazarsfeld, Victor Lozovanu, and Damien Roy for helpful discussions. We are also extremely grateful to the referees of this paper for pointing out several mathematical and expositional errors in the initial versions, and for their suggestions on how to correct them. Finally, we wish to acknowledge an intellectual debt to Michael Nakamaye who has long advocated the point of view that Seshadri constants are diophantine.

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Correspondence to Mike Roth.

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David McKinnon was partially supported by an NSERC research grant. Mike Roth was partially supported by an NSERC research grant.

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McKinnon, D., Roth, M. Seshadri constants, diophantine approximation, and Roth’s theorem for arbitrary varieties. Invent. math. 200, 513–583 (2015). https://doi.org/10.1007/s00222-014-0540-1

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Mathematics Subject Classification

  • Primary 14G05
  • Secondary 14G40