Ok, this is going to be my last post in enumerative geometry for a while, as I’m kind of drifting away from the subject. However, this one will be fun. We’ve already established the structure of the cohomology ring for Grassmannians, so what we’re going to do is talk about what the word “generic” means in some very precise contexts, and then we’re going to count a bunch of things.
So first and simplest, what does it mean to be a generic line in ? That should mean something like “for most lines” and “if you change the line a little, the same properties hold.” So how do we make that rigorous? Well, we look at the Grassmannian of lines in , . We’ll say that a property holds generically if it is true for a Zariski open subset of .
More generally, if we have any sort of object that we want to prove a statement about (for instance, hypersurfaces of given degree, curves of a given genus, linear subspaces of projective space, complete intersections, etc), we say that a property holds generically or that it is true at a general point if it is true for a Zariski open subset of the moduli space of such things. In slightly worse situations, we say that a property holds at a very general point if it is true for a countable intersection of nonempty Zariski open subsets. We won’t need this notion, but it’s out there. Everything we talk about should be merely generic.
So for our first Schubert calculus trick, we take four lines in general position in . That just means that there is some Zariski open subset of on which our statement will hold. We want to know how many lines intersect all four of them. We know from before that the variety of lines intersecting a given lines has cohomology class , and that the cohomology class is independent of the actual flag chosen. So we want the number to be , which is twice the class of a point on .
Now, there’s a general principle which we must invoke. For our purposes, we’ll state it informally as any Schubert calculation works generically. For this particular problem, it says that there’s an open set on the moduli space of four lines such that we get two distinct points on the Grassmannian, rather than a single line counted twice, and no excess intersection (which we’ll not talk about).
Problems like that one are the simplest example of an application of Schubert Calculus. All we’re doing is solving problems involving linear subspaces of . While nifty, if this were all it could do, it wouldn’t be nearly as interesting. So now we’ll move on to my favorite application, which I’ve mentioned before. We can use Schubert calculus to talk about the collection of all lines in a general hypersurface, and in fact, can generalize it to complete intersections.
Now, look at , the Grassmannian of lines in . This comes equipped with a natural rank 2 vector bundle, , which is naturally a quotient of the trivial rank 4 bundle as the quotient of representing the line.
Now, say we want to study the family of lines in a degree hypersurface in . Then we look at on the Grassmannian. Now, this essentially amounts to taking a generic polynomial of degree , and restricting it to each line. So we want to look at where this polynomial restricts to zero. All in all, that means we need to zero section of , which will be its top Chern class. This cohomology class will be the class of the variety of lines in a general hypersurface of degree . Now, to make it work for complete intersections, we just look at the varieties of lines in each hypersurface and intersect them, so we multiply the top Chern classes together.
So what Schubert calculus will let us do is work out the cohomology class of the variety of linear spaces of a given dimension contained in any complete intersection. This is very useful, and gives a starting point for other, more sophisticated methods of enumerative geometry, like Gromov-Witten theory and Quantum Cohomology (which tend to use Schubert stuff for initial conditions in recurrences).