Quartic K3 surfaces without nontrivial automorphisms

  • Date: 02/01/2007

Ronald van Luijk (PIMS, SFU, UBC)


University of British Columbia


We will deal with a gap in a result of Bjorn Poonen. He found explicit
examples of hypersurfaces of degree d³3 and dimension n³1 over any
field, such that the group of automorphisms over the algebraic closure
is trivial, except for the pairs (n,d)=(1,3) or (2,4). Examples of the
former pair, cubic curves, do not exist. We deal with the remaining
case, quartic surfaces. For any field k of characteristic at most 19 we
exhibit an explicit smooth quartic surface in projective threespace
over k with trivial automorphism group over the algebraic closure of k.
We also show how this can be extended to higher characteristics. Over
the rationals we also construct an example on which the set of rational
points is Zariski dense.

Other Information: 

SFU/UBC Number Theory Seminar 2007