Seshadri constants, Diophantine approximation, and Roth's Theorem for arbitrary varieties

David McKinnon; Mike Roth

doi:10.1007/s00222-014-0540-1

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arXiv · 1306.2976

Seshadri constants, Diophantine approximation, and Roth's Theorem for arbitrary varieties

David McKinnon·Mike Roth

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Abstract

In this paper, we associate an invariant $α_{x}(L)$ to an algebraic point $x$ on an algebraic variety $X$ with an ample line bundle $L$. The invariant $α$ 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 $ε_{x}(L)$ measuring local positivity of $L$ at $x$, and in particular that Roth's theorem on $\mathbf{P}^1$ generalizes as an inequality between these two invariants valid for arbitrary projective varieties.

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BibTeX RIS

David McKinnon, Mike Roth. 2015-04-27. Seshadri constants, Diophantine approximation, and Roth's Theorem for arbitrary varieties. https://doi.org/10.1007/s00222-014-0540-1

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