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

• Douglas Ulmer ^[1] ; Giancarlo Urzúa ^[2]
  1. [1] University of Arizona

     University of Arizona

     Estados Unidos

     
  2. [2] Pontificia Universidad Católica de Chile

     Pontificia Universidad Católica de Chile

     Santiago, Chile

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 28, Nº. 2, 2022
• Idioma: inglés
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• Resumen

  • 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.