## 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.

SFU/UBC Number Theory Seminar 2007