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Transversality of sections on elliptic surfaces with applications to elliptic divisibility sequences and geography of surfaces

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Abstract

We consider elliptic surfaces \({\mathcal {E}}\) over a field k equipped with zero section O and another section P of infinite order. If k has characteristic zero, we show there are only finitely many points where O is tangent to a multiple of P. Equivalently, there is a finite list of integers such that if n is not divisible by any of them, then nP is not tangent to O. Such tangencies can be interpreted as unlikely intersections. If k has characteristic zero or \(p>3\) and \({\mathcal {E}}\) is very general, then we show there are no tangencies between O and nP. We apply these results to square-freeness of elliptic divisibility sequences and to geography of surfaces. In particular, we construct mildly singular surfaces of arbitrary fixed geometric genus with K ample and \(K^2\) unbounded.

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Authors and Affiliations

  1. Department of Mathematics, University of Arizona, Tucson, AZ, 85721, USA

    Douglas Ulmer

  2. Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Santiago, Chile

    Giancarlo Urzúa

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  1. Douglas Ulmer
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  2. Giancarlo Urzúa
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Correspondence to Douglas Ulmer.

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Ulmer, D., Urzúa, G. Transversality of sections on elliptic surfaces with applications to elliptic divisibility sequences and geography of surfaces. Sel. Math. New Ser. 28, 25 (2022). https://doi.org/10.1007/s00029-021-00747-x

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