Soolin describes the planet Doma as being in the Sixth Quadrant.
A Quadrant is a Quarter (The clue is in the name!) so there cannot be a sixth quarter.
At the auction there is a servant using a large feather-covered cooling fan on a stick. However, there is also a stiff breeze blowing, rendering any fanning action useless.
Avon, Tarrant, and Soolin just had a big discussion with Piri about why all three of them had to go look for Cancer. Having Soolin sit down to think rather than actively searching makes absolutely no sense whatsoever, except that it is a ridiculous way to set up Cancer trying to kill her with the robot spider.
There is no reason for Nebrox to leave the Scorpio to go to Cancer's ship with Vila and Soolin.