I was talking to this fellow mathematician, and told him how for my very personal taste, the research in algebraic geometry has sailed a bit too far ashore from the original question that motivated the subject: A classification for the solutions of polynomial equations. I told him that most of the heavy machinery in algebraic geometry can be given very good intuition - however, most courses neglect to do so.
Anyways, he was clearly amused and not quite in belief of my radical position that unconditional and solitary abstraction isn't the only way to do algebraic geometry. He teased me to then tell him what an *»ample«* line bundle was, without talking about tensor powers or commutative diagrams.
So, that's the reason for this post. I answered that ample line bundles are precisely those that induce finite morphisms to projective space, but I couldn't remember where I knew it from. I searched through all the major literature and could not find a reference until someone on Mathoverflow helped me out. Since I complained so much about how noone ever gives intuition to these kinds of concepts in algebraic geometry, I decided I'd blog a bit about line bundles and how you should think about them.
Do you want to know more?