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

Douglas Ulmer, Giancarlo Urzúa

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

We consider elliptic surfaces 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 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 K2 unbounded.

Original languageEnglish (US)
Article number25
JournalSelecta Mathematica, New Series
Volume28
Issue number2
DOIs
StatePublished - May 2022

Keywords

  • Elliptic divisibility sequences
  • Elliptic surfaces
  • Geography of surfaces
  • Stable surfaces
  • Unlikely intersections

ASJC Scopus subject areas

  • Mathematics(all)
  • Physics and Astronomy(all)

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