TITLE: Morphisms of (quasi-)projective varieties
QUESTION [14 upvotes]: 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?

REPLY [2 votes]: It is impossible to give a concrete description of a map between two abstract varieties without first giving some concrete way of representing those varieties.  That is why Hartshorne’s definition of morphism is in terms of abstract properties. If you tell how the varieties are represented, then it is possible to describe a morphism in those same terms.  To check Hartshorne's definition, you must have a way of representing both the morphism and the regular rational functions.  Thus concrete descriptions of maps of projective varieties must use projective models.  This answer is hence in the spirit of those of David and Charles.
Examples:

The first step in visualizing an algebraic morphism of a projective variety is visualizing an algebraically embedded set, e.g. a plane curve C of degree d.  E.g. imagine a cubic in the plane, or any curve of degree d.  Then an example of a morphism defined on that curve is projection π of the curve C from a point p to a line L.

For each point x of the curve C, π(x) is the intersection of the line through p and x with L.  The fiber π^-1(y) of π over a point y of L is the intersection of the line through p and y with the curve.  Thus fibers of π correspond to lines through p.  This suggests that the number of points on different fibers of π is a constant equal to d, and that the fibers vary in a “linear family”.   Lines through p tangent to C meet C in fewer than d points, so π represents C as a “branched cover” of degree d over L, which is a covering space except over images of those tangential intersections.  
Algebraically we are taking a polynomial in x and y, fixing y, and representing the fiber over y as the solutions of the resulting polynomial in x.  That polynomial generically has degree d in x, and branching occurs when it has multiple roots.  Rational functions of y which are regular at y0 pull back to (the same) rational functions on C, regular at points (x,y0) over y0, as in Hartshorne’s definition of morphism.

To define a map f from the plane curve C into 3 space P^3, one can choose a linear family spanned by 4 auxiliary plane curves more general than lines, such as any 4 independent conics passing through two specific points p,q of the plane not on C.  This will map the entire plane, except the two points p,q where the map f is not defined, to a surface Q in P^3, and maps C onto this surface.  Since the linear coordinate functions on P^3 pull back under f to the quadratic polynomials defining the family of plane conics, a plane in P^3 meets f(C) in a set of points which pull back to the 6 common points of the cubic C with one of these conics.  Thus f(C) is a sextic curve in P^3.   Rational functions a/b on P^3 which are regular at f(x) pull back via substitution to rational functions a(f)/b(f) on C which are regular at x.  Indeed all morphisms of plane curves can be given by rational functions.
Given a surface S of degree d, and a line L in P^3, one can define an algebraic projection map π from S to L by choosing an auxiliary line M, and considering all planes passing through M.  Given a point x of S, π(x) = the point where L meets the plane spanned by x and M.  Then for a point y of L, the fiber π^-1(y) is the intersection curve of S with the plane spanned by y and M.  As this family of planes revolves around M, they sweep out on S the pencil of curve fibers of π.  Since the line L also meets S in a finite set of d points, the map π is not defined at those d points.  Except for these base points, an algebraic map of S to a line again amounts to fibering the source S into a linear family of algebraic subsets of codimension one, i.e. into a linear family of “divisors”.  If L is the z axis, and S is defined by a polynomial G(x,y,z) = 0, the fiber π^-1(z) is the plane curve of solutions of G=0 in x,y obtained by fixing z.  Again rational functions of z regular at π(p), pull back to rational functions regular at p on S, away from the d points common to L and S.

Remark: It helps to recall that Hartshorne actually taught and wrote the chapters on curves and surfaces before the abstract chapters.  He indeed gives a very nice description of projective morphisms of curves on pages 309-314, and the later discussion of cubic surfaces is equally concrete.  Indeed every morphism from one projective variety to another is obtained as in these examples, first mapping into a high dimensional projective space by rational functions, then projecting down to a lower one.