Transversality of sections on elliptic surfaces with applications to elliptic divisibility sequences and geography of surfaces
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Date
2022
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Abstract
We consider elliptic surfaces epsilon 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 epsilon 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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Elliptic surfaces, Unlikely intersections, Elliptic divisibility sequences, Stable surfaces, Geography of surfaces