Morphisms of (quasi-)projective varieties

Question

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:

1. 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?

2. 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?

3. 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?

Answer 1

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 U_i and V_i of X and Y such that f takes U_i to V_i and f:U_i --> V_i 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, ..., f_n) 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.

Answer 2

I think a good analogy here is differentiable functions.

Morphism from a quasi-projective variety to the affine line is just the regular functions, i.e. a bunch of compatible rational functions. Compare this to differentiable functions from a smooth manifold M to R.

Next, morphism from a quasi-projective variety to the affine space $\mathbb{A}^n$ is just n regular functions. Compare this to differentiable functions from a smooth manifold to R^n. Then we know how to define morphism from a quasi-projective variety to an affine variety: just embed affine variety into A^n, and use the previous definition. Compare this to differentiable functions from smooth manifold to a submanifold of R^n.

Finally, morphism $f$ from a quasi-projective variety to another variety can be defined by choosing an open affine cover $U_i$, and require all the maps $f^{-1}(U_i) \rightarrow U_i$ are morphisms. Compare this to differentiable functions from smooth manifold to another.

This definition, though more concrete, is not good when we deal with schemes, because it depends much on embedding. So in general, consider a morphism as a sheaf map is much better. Again, this is parallel to the situation of manifolds. You can regard a smooth manifold as a locally ringed space over R which is locally isomorphic to sheaf of smooth functions on some open subset of R^n. Then the usual notion of differentiable maps between manifolds would correspond to our situation.

Answer 3

Hopefully I can go some of the way toward addressing 2 and 3 without getting carried away and getting too technical/newbie unfriendly, although I have a feeling I'm going to fail at this last part.

The answer to 3 is that yes, there are ways of making the notion of morphism of varieties (or more generally schemes) intuitively obvious (at least for me). For instance one can view a variety as a certain type of locally ringed space. A locally ringed space consists of a topological space, in this case the underlying set of points of the variety with the Zariski topology, together with a sheaf of rings which is just a way of keeping track of the regular functions on the various open subsets and how one can glue them together (plus the locally part which comes down to the fact that functions not vanishing at a point are invertible in a neighbourhood of that point). From this point of view a morphism of varieties is precisely what it has to be - a continuous map of topological spaces together with a map of sheaves which tells you what happens to regular functions when they are pulled back. This is really just a restatement of the definition you know, but from the point of view of sheaf theory there is really only one thing a morphism of sheaves can be. (One can see it "is what it has to be" via a functor of points interpretation or by gluing as Simon outlines in his answer)

As far as 2 goes two examples come to mind (although I guess this is still a sort of scheme theoretic rather than classical variety type approach). The first is that over an affine base any projective scheme comes from a graded ring via the Proj construction. Using this one can construct certain morphisms from quasi-projective varieties to projective ones from graded morphisms of graded rings in analogy with the way one gets morphisms of affine varieties from morphisms of rings.

The second comes from the adjunction between Spec and global sections - i.e. for any ring A (commutative with unit) and any variety (this works for schemes in general) X there are natural bijections
Hom_{Schemes}(X, Spec A) ~ Hom_{Rings}(A, Γ(X,O_X))
There is a special case of this which I think is informative (at least it is a useful thing to know). Suppose we are working over some fixed base field k which is algebraically closed (although this is fine more generally as well) and consider maps from X to Spec k[t] (the affine line over k). Then this bijection tells us that to give such a map is the same thing as giving a globally defined regular function on X (the global section which we send the indeterminate t to). So if one knows the globally defined regular functions on X then one knows all of the morphisms to the affine line. Conversely one can actually use this interpretation to show that the only globally defined regular functions on a projective scheme are constants (although this is probably overkill).

I hope that this is at least vaguely helpful!

Answer 4

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:

1. 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.

1. 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.

2. 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.

Answer 5

As you imply that you know, for affine varieties a morphism is just a map that comes from a ring homomorphism between the associated rings. ie there is a correspondance between morphisms Spec(A) to Spec(B) and ring homomorphisms B to A.

A general variety is covered by affine varieties, and a morphism between two such should locally be a morphism betweeen affine varities. That is, to define a morphism for V to W is to choose an open cover of V by affines and then write down morphisms on each open set in your cover that land in an affine subvariety of W in such a way that they agree on all interestions in your cover. You can think of this an analogous to analytic continuation in complex analysis, or defining a map between manifolds by defining it on charts.

Answer 6

The definition in Hartshorne is, surprisingly, the most natural one.

Let's approach this in a following way. If you meditate about what Grothendickian algebraic geometry is, you grok suddenly that there it tells a story of some forms that are desribed algebraically but imagined geometrically. The first such form is A^1, affine line. Take a time to focus on both the techniques (meditate on k[x]) and content (a geometric qi, if you wish) related to affine line.

Now let's go to the form called P^1. You should be able to imagine P^1 — it's simply a sphere, or a space of lines on a plane or whatever image you like. Now, what's the technique that would describe this formally? If you think about it, you'll come up with the same idea — it's something made out of two simpler things, the A^1s.

Ok, now lets think about maps between these forms. E.g. take arbitrary point-to-point mapping. It's clearly something wrong, not beautiful, because I've thrown away all the intuition I've asked you to create. So, let's concentrate better. Our structure is algebraic. For an affine manifold, this has been formalized as "polynomials go to polynominals". For a smooth manifold it's been formalized as "locally, smooth functions go to smooth functions".

Now, if we combine so, we get the definition of "locally, polynomials go to polynomials" — and this is the definition that is useful in practice. The Hartshorne simply has it written in an abstract language that can be applied to any scheme. That who masters this example shall have no fear.

Answer 7

Ok. Concrete. A morphism of projective (quasi-projective) varieties f:V->W can be described by taking V and W, embedding them into projective spaces, and then f will be a collection of homogeneous rational functions, all of the same degree, such that the image of V is contained in W. The intuition here is that we're looking at algebraic maps of algebraic objects. Naturally, this is a bad way to do things from a modern perspective, but you said you wanted concrete. So as for motivating examples, take any rationally parameterized anything in projective space, and it's a morphism from some open subset of another projective space (where the rational functions are all defined).