On an Enriques surface associated with a quartic Hessian surface

Ichiro Shimada

Published 2017-01-03Version 1

We present several algorithmic methods to investigate the geometry of a complex Enriques surface by computation on the N\'eron-Severi lattices of the Enriques surface and of its covering K3 surface. We apply them to an Enriques surface Y whose \'etale double cover X is birational to a general quartic Hessian surface. Using the result on the automorphism group of X due to Dolgachev and Keum, we obtain a finite presentation of the automorphism group of Y. This automorphism group is compared with the automorphism group of the generic Enriques surface. A fundamental domain of the action of the automorphism group on the nef cone of Y is described explicitly. The list of elliptic fibrations on Y and the list of combinations of rational double points that can appear on a surface birational to Y are presented.