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.