Today, we’re going to talk about a very important class of rational varieties, that show up all the time, in quite a variety of different contexts, and at the end, we’ll talk about why.
Let and . Then, take complementary of dimensions and , that is, , and their span is all of . Now, choose and rational normal curves. Finally, choose an isomorphism .
A (2d) rational normal scroll is . That is, it’s the union of the lines from one rational normal curve to another. This depends only on the numbers , and not on the choice of subspaces, rational normal curves, or isomorphism.
Now, if , then the lines used in the definition are the only lines on , and we’ll call those the lines of the ruling. However, for , we can get some classical examples that we’ve seen before. is given by taking two skew lines in , and takes the unions of lines between them, so this gives a quadric surface. The next simples, , turns out to be what you get when you embed the plane into as the Veronese surface, and then project to from a point on the surface.
We can generalize the construction even further: set such that , and pick complementary subspaces. Next, pick rational normal curves, and isomorphisms. Then the -dimensional rational normal scroll is . This is also called the rational normal -fold scroll.
Two quick examples of these are that is the Segre embedding of into , and more generally, times is the Segre of into .
So, now that we’ve talked about rational normal scrolls, the following theorem can be proved:
Theoem: Let be an irreducible and nondegenerate variety of dimension . Then the minimum possible degree of is and the possible varieties with this degree are:
- Quadric hypersurfaces
- The cone over the quadratic Veronese
- Rational normal scrolls
This isn’t trivial to prove, but is VERY useful. Here are a few consequences:
- If is a rational normal -fold scroll, then a hyperplane section is a -fold scroll.
- Projection from a point of a scroll is a scroll.
- The examples above actually can be proved from this theorem
- Rational normal curves are minimal curves
There are quite a few other consequences of this, and this is connected to classical Castelnuovo Theory, and to some much more recent work of Pareschi and Popa, generalizing this to abelian varieties.