This is another "homework help" question, which is still hopefully of at least pedagogical interest to working mathematicians.
So, I'm currently taking an intro algebraic geometry class, and one thing I've had some trouble with is grokking what a morphism of projective or quasi-projective varieties should be. I know at least one definition by heart, which is Hartshorne's, which is the one about locally pulling back regular functions to regular functions.
The problem is, I don't have the Grothendieckian superpower of being able to grasp these abstract ideas without playing around with some concrete examples. And Hartshorne's definition isn't all that conducive to actually checking in practice whether a given map is actually a morphism. So my question has, I guess, three parts:
Is there a more concrete definition of a morphism of projective/quasi-projective varieties that I can use in practice to check if something's a morphism?
What are some of the motivating examples of morphisms between varieties, that give one a sense of what they should be? What's an example of something that isn't a morphism, that gives one a sense of what's too much to ask?
Most abstractly, is there a big-picture explanation that makes the definition of morphism "intuitively obvious," as is the case (for instance) for groups, or even for affine varieties?