Dr. Josh Keyes: So what's this about then?
FBI Agent: We don't know. You have higher security clearance than us.
Dr. Josh Keyes: I have security clearance?
FBI Agent: Yes sir, we're just here to take you to your jet.
Dr. Josh Keyes: I have a jet?!
The geodes occurring in the mantle during the film are impossible in two ways. Firstly, there are no gaps at those depths. The pressure is approximately 3.5 million times surface pressure, and it is not feasible that such structures could form, let alone be maintained. Secondly, the crystals inside the geodes are described as amethysts. Amethysts are a purple variety of quartz, and as any undergrad geology student could tell you, there is no quartz in the mantle, it simply is not stable at such high temperatures.
If you look closely with time-frame advance during the pigeon scene you will see a fish "flying" into a window instead of a pigeon.