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 {k\leq \ell} and {n=k+\ell+1}. Then, take {\Lambda,\Lambda'\subset \mathbb{P}^n} complementary of dimensions {k} and {\ell}, that is, {\Lambda\cap\Lambda'=\emptyset}, and their span is all of {\mathbb{P}^n}. Now, choose {C\subset\Lambda} and {C'\subset\Lambda'} rational normal curves. Finally, choose an isomorphism {\phi:C'\rightarrow C}.

A (2d) rational normal scroll is {S_{k,\ell}=\cup_{p\in C'} \overline{p,\phi(p)}}. That is, it’s the union of the lines from one rational normal curve to another. This depends only on the numbers {k,\ell}, and not on the choice of subspaces, rational normal curves, or isomorphism.

Now, if {k\neq 1}, then the lines used in the definition are the only lines on {S_{k,\ell}}, and we’ll call those the lines of the ruling. However, for {k=1}, we can get some classical examples that we’ve seen before. {S_{1,1}} is given by taking two skew lines in {\mathbb{P}^3}, and takes the unions of lines between them, so this gives a quadric surface. The next simples, {S_{1,2}}, turns out to be what you get when you embed the plane into {\mathbb{P}^5} as the Veronese surface, and then project to {\mathbb{P}^4} from a point on the surface.

We can generalize the construction even further: set {a_1\leq\ldots\leq a_k} such that {\sum a_i=n-k+1}, and pick {\Lambda_i\cong \mathbb{P}^{a_i}\subset\mathbb{P}^n} complementary subspaces. Next, pick {C_i\subset\Lambda_i} rational normal curves, and {\phi_i:C_1\rightarrow C_i} isomorphisms. Then the {k}-dimensional rational normal scroll is {S_{a_1,\ldots,a_k}=\cup_{p\in C_1}\overline{p,\phi_2(p),\ldots,\phi_k(p)}}. This is also called the rational normal {k}-fold scroll.

Two quick examples of these are that {S_{1,1,1}} is the Segre embedding of {\mathbb{P}^2\times \mathbb{P}^1} into {\mathbb{P}^5}, and more generally, {S_{1,\ldots,1}} {k} times is the Segre of {\mathbb{P}^{k-1}\times \mathbb{P}^1} into {\mathbb{P}^{2k-1}}.

So, now that we’ve talked about rational normal scrolls, the following theorem can be proved:

Theoem: Let X\subset\mathbb{P}^n be an irreducible and nondegenerate variety of dimension k.  Then the minimum possible degree of X is n-k+1 and the possible varieties with this degree are:

  1. Quadric hypersurfaces
  2. The cone over the quadratic Veronese v_2(\mathbb{P}^2)\subset\mathbb{P}^5
  3. Rational normal scrolls

This isn’t trivial to prove, but is VERY useful.  Here are a few consequences:

  • If S\subset\mathbb{P}^n is a rational normal k-fold scroll, then a hyperplane section is a (k-1)-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.

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