In algebraic geometry, we like to make statements like: “two conics meet at points”, “a degree four plane curve has bitangents”, “given four lines in three space, there are lines that meet all of them”. In each of these, we are saying that, as some parameter (the conics, the degree four curve, the lines) changes, the number of solutions to some equation stays constant. The “principle of conservation of number” refers to various theorems which make this precise.
In my experience, students in algebraic geometry tend to pick up the rough idea but remain hazy on the details, most likely because there are many different ways to make these details precise. I decided to try and write down all the basic results I could think of along these lines.
Let be some parameter space such as the space of pairs of two conics. Let be some space of solutions, such as the space of triples where is a point on . Let be a map, such as projection onto the components. We want theorems which will discuss the size of the fibers of , in terms of some global degree of the map .
We work over some field . For simplicity of presentation, we’ll assume that is affine, meaning that it is a subset of defined by polynomial equations
We’ll write for the ring .
It would be silly to ask for any such results if were disconnected. A very basic observation of algebraic geometry is that is connected if and only if has no nontrivial idempotents. In fact, we will ask for something stronger: That is an integral domain. The terminology for this is that is irreducible. From now on, we will make:
Assumption is irreducible. ( is an integral domain.)
If is also affine, with corresponding ring , then is an module. We define the degree of in this case to be the dimension of as a vector space. Degree can be defined in much greater generality; we will feel free to refer to it in greater generality without giving the definition. We will denote the degree of by . Roughly, we want theorems which say that the fibers of have size .
Here is our first result.
Theorem (Shafarevich, II.6.3, Theorem 4) If has characteristic zero and is algebraically closed then for almost all in . More precisely, there is some polynomial , not identically zero on , so that implies .
Warning This isn’t true if is not algebraically closed: Consider the map from .
Warning This isn’t true in characteristic : Consider .
We now want results which let us say something, not just about almost all , but about all .
We will at first focus on counting the size of in a naive sense: We think of as sitting in (or in ) and we literally count points of the fiber. We can’t hope for the fibers to always be of full size because even the nicest map, , has fiber of size , not , over the point . So, using the naive size, we can only hope for upper bounds.
There are two additional problems. The first one is if we have something like projecting onto the coordinate. In this case, the degree is but the fiber over has size . When is affine, with corresponding ring , we can fix this by requiring that is torsion free as an -module. In general, the right condition is that no irreducible component of maps to a proper subvariety of .
More subtly, suppose that is a nodal curve, such as , and is its desingularization. (In this case, the line with as the map .) Then the degree of the map is , but the fiber over is , of size . The hypothesis to rule this out is that is integrally closed in its fraction field. By definition, this is the same as saying that is normal.
Once we rule out these possibilities, we have
Theorem (Shafarevich, II.6.3, Theorem 3) If is normal, and no irreducible component of maps to a proper subvariety of , then every fiber of has naive size .
I can’t resist mentioning a result which far harder than these:
Theorem (A consequence of Zariski’s Main Theorem) Let be normal and let have degree . Assume that no irreducible component of maps to a proper subvariety of . For any in , the number of connected components of is at most $d$.
Scheme theoretic size
We now consider counting size in a less naive way. Again, for simplicity, suppose that is affine, with corresponding ring . Let be a point of , so there is a map of rings by . Consider the ring , where acts on by the above map. The maps from this ring to are the point in . Thus, is an upper bound for the number of points of above . We will call this dimension the scheme theoretic size of the fiber. Once again, it can be defined when is not affine as well.
We have the following cautionary example: Let mapping onto the coordinate. Then the degree is , but the fiber above has size , either scheme theoretically or naively. To rule this out, we impose that is finite over . By definition, this means that is affine, and is a finitely generated module.
You might worry about how we could ever prove that is affine if it is not given to us as a closed subset of . Fortunately, we have:
Theorem (Hartshorne, Exercise III.11.2) If is projective with finite fibers, then it is a finite map. Here projective means that is a closed subset of , projecting onto . (This is not the morally right definition of a projective map, but if you are ready for the right definition, then you should be working with “proper” rather than “projective” anyway.)
We then have
Theorem (Hartshorne, Exercise II.5.8) If is finite over , and no irreducible component of maps to a proper subvariety of , then every fiber of has scheme theoretic size .
Theorem Let be a finite map. Then all fibers have scheme theoretic size if and only if is flat over .
Unfortunately, flat is a rather technical condition. The first thing to understand is that some nice looking maps can fail to be flat:
Warning Let be , let and let the map be . This is a finite map. (We can alternately describe as .) This map is degree , but the fiber over has scheme theoretic size (and naive size ).
If your eye is well enough trained that this doesn’t look nice to you, try the examples here.
There are two good conditions that imply flatness:
Theorem (Hartshorne III.9.7) If is normal and one dimensional, and no irreducible component of maps to a proper subvariety of , then is flat over .
Theorem (The miracle flatness theorem) If is Cohen-Macaulay, is smooth of the same dimension as , and is finite, then is flat.