Rational Normal Scrolls

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:

Quadric hypersurfaces
The cone over the quadratic Veronese $v_2(\mathbb{P}^2)\subset\mathbb{P}^5$
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.

About Charles Siegel

Charles Siegel is currently a postdoc at Kavli IPMU in Japan. He works on the geometry of the moduli space of curves.

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