TY - JOUR

T1 - Transversality of sections on elliptic surfaces with applications to elliptic divisibility sequences and geography of surfaces

AU - Ulmer, Douglas

AU - Urzúa, Giancarlo

N1 - Funding Information:
The first-named author thanks Seoyoung Kim, Nicole Looper, and Joe Silverman for helpful conversations at the 2019 AMS Mathematics Research Community meeting in Whispering Pines, Rhode Island, and for pointing out [] and its antecedents. He also thanks the Simons Foundation for partial support in the form of Collaboration Grant 359573. The second-named author thanks FONDECYT for support from grant 1190066. Both authors thank Matthias Schütt for comments and corrections, and they thank Pietro Corvaja, Brian Lawrence, and Umberto Zannier for their comments on an earlier version of this paper and their pointers to related literature, notably the preprint [].
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

PY - 2022/5

Y1 - 2022/5

N2 - 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.

AB - 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.

KW - Elliptic divisibility sequences

KW - Elliptic surfaces

KW - Geography of surfaces

KW - Stable surfaces

KW - Unlikely intersections

UR - http://www.scopus.com/inward/record.url?scp=85122062483&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85122062483&partnerID=8YFLogxK

U2 - 10.1007/s00029-021-00747-x

DO - 10.1007/s00029-021-00747-x

M3 - Article

AN - SCOPUS:85122062483

VL - 28

JO - Selecta Mathematica, New Series

JF - Selecta Mathematica, New Series

SN - 1022-1824

IS - 2

M1 - 25

ER -