## 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:

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.