I recently started studying algebraic geometry together with a couple of friends and especially in discussions online we keep reading about developing geometric intuition. There are some questions on this website about developing geometric intuition, but none of them really ask for what geometric intuition is. It doesn't seem obvious what the whole concept of geometric intuition means in the context of modern algebraic geometry, so it seems hard to judge when one has started developing it. Especially, when problems in number theory can often be related to algebraic geometry and then solve by using this geometric intuition.
Let me give an example from elementary analysis, where this is completely obvious. Take the squeeze theorem. Anyone can visualize two graphs in their head and "see" that anything between them must get pushed to the same point. Then the proof just corresponds to having learned how to translate a picture to a formal epsilon-delta argument. My question is then that is an algebraic geometer or arithmetic algebraic geometer working actually seeing nice pictures of lines, surfaces and curves in their head? Or is it just a matter of having developed experience with how different algebraic objects behave? The latter wouldn't seem any different from having developed intuition about, say, field theory through experience and this intuition could hardly be called "geometric" by anyone.
Feel free to close if this question is considered inappropriate for this website, I certainly understand. The reason for posting here instead of math.stackexchange is that graduate students in my department don't seem to have the experience themselves to answer it. I'm sure that this intuition keeps developing for a long time after finishing graduate studies. Hence, I was hoping for an experienced audience hopefully willing to answer.