Here I give the solution to my own problem.
For my desktop PC I have deactivated suspend and hibernate. That seems to prevent GDM from blanking the screen when idle. The solution I found is pretty easy:
With dconf-editor I changed the values of the following keys:
org.gnome.settings-daemon.plugins.power sleep-inactive-ac-type
org.gnome.settings-daemon.plugins.power sleep-inactive-battery-type
to “blank” instead of “suspend”.
That did the trick.