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:
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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?
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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?
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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?
Best Answer
You say that you are comfortable with morphisms of affine varieties. A map f: X --> Y between quasi-projective varieties is a morphism if and only if we can give open affine covers Ui and Vi of X and Y such that f takes Ui to Vi and f:Ui --> Vi is a morphism.
Conceptually, I think of a morphism as anything I can write down in terms of polynomials without using limits or cases. The only case that used to trip me up is things involving normaliztion. For example, if Y is Spec k[x,y]/y^2-x^3 and X is Spec k[t], the map from Y --> X by (x,y) --> y/x is NOT a morphism, even though it has a well defined limit at (0,0). And writing it as (x,y) --> y/x if (x,y) \neq (0,0) and --> 0 if (x,y)==0 also doesn't work; because it uses cases. On the other hand, the map X --> Y by t --> (t^2, t^3) is perfectly good.
Of course, this was an affine example. But a general morphism is just a map which is locally an affine morphism; so it is a map which I can locally write in terms of polynomials.
In the particular case of projective varieties, if X \subset P^m and Y \subset P^n, and (f0,f1, ..., fn) are homogenous polynomials of the same degree, with no common zero on X, and such that these polynomials take X to Y, then these polynomials define a morphism. Note that this is not if and only if; this theorem is for writing down examples of morphisms, not defining them. I think there may be a way to make this if and only if, by allowing you to change the projective embeddings, but I'm not sure of the details.