In a projective space $\mathbb{P}^{k+2}$ consider two complementary subspaces $\mathbb{P}^1,\mathbb{P}^k$, and let $C\subset\mathbb{P}^k$ be a degree $k$ rational normal curve. Fixed an isomorphism $\phi:\mathbb{P}^1\rightarrow C$ we consider the rational normal scroll


$$S\_{(1,k)} = \bigcup\_{p\in \mathbb{P}^1}\left\langle p, \phi(p)\right\rangle\subset \mathbb{P}^{k+2}$$


where $\left\langle p, \phi(p)\right\rangle$ is the line through $p$ and $\phi(p)$. 


Now, let $\Sigma\_{(1,k)}$ be the Segre embedding of $\mathbb{P}^1\times \mathbb{P}^k$ in $\mathbb{P}^{2k+1}$.


Can $S\_{(1,k)}$ be recovered intersecting $\Sigma\_{(1,k)}$ with a linear subspace of dimension $k+2$ of $\mathbb{P}^{2k+1}$? This is known to be true if $k = 2$. Indeed a general hyperplane section of $\Sigma\_{(1,2)}\subset\mathbb{P}^5$ is a scroll surface of the form $S\_{(1,2)}$.



Linear sections of Segre varieties and rational normal scrolls
The scroll $S\_{(1,k)}$ can be locally parametrized by the map
$$
\begin{array}{cccc}
\phi: & \mathbb{A}^1\times\mathbb{P}^1 & \longrightarrow & \mathbb{P}^{k+2}\\ 
 & (u,[\alpha\_0:\alpha\_1]) & \mapsto & [\alpha\_0 u:\alpha\_0:\alpha\_1 u^k:\alpha\_1 u^{k-1}:\dots:\alpha\_1 u:\alpha\_1].
\end{array} 
$$ 
Now, consider the Segre embedding 
$$
\begin{array}{cccc}
\sigma: & \mathbb{P}^1\times\mathbb{P}^k & \longrightarrow & \mathbb{P}^{2k+1}\\ 
 & ([u:v],[\alpha\_0:\dots:\alpha\_k]) & \mapsto & [\alpha\_0u:\dots:\alpha\_ku:\alpha\_0v:\dots :\alpha\_kv].
\end{array} 
$$ 
and let $\Sigma\_{(1,k)}$ be its image. Note that $\Sigma\_{(1,k)}$ is locally parametrized by
$$
\begin{array}{cccc}
\widetilde{\sigma}: & \mathbb{A}^1\times\mathbb{P}^k & \longrightarrow & \mathbb{P}^{2k+1}\\ 
 & ([u:1],[\alpha\_0:\dots:\alpha\_k]) & \mapsto & [\alpha\_0u:\dots:\alpha\_ku:\alpha\_0:\dots :\alpha\_k].
\end{array} 
$$ 
and that $\deg(\Sigma\_{(1,k)}) = \deg(S\_{(1,k)}) = k+1$. Now, take $\alpha\_{i} = \alpha\_1u^{i-1}$ for $i=2,\dots,k$. Then
$$\widetilde{\sigma}(u,[\alpha\_0:\alpha\_1:\alpha\_1u:\dots :\alpha\_1u^{k-1}]) = [\alpha\_0u:\alpha\_1u:\alpha\_1u^2:\dots:\alpha\_1u^{k-1}:\alpha\_1u^k:\alpha\_0:\alpha\_1:\alpha\_1u:\dots\alpha\_1u^{k-1}]$$
and the coordinates functions of this last map are exactly the ones appearing in the above expression of $\phi$. 


Therefore, if $[Z\_0:\dots:Z\_{2k+1}]$ are the homogeneous coordinates on $\mathbb{P}^{2k+1}$ and 
$$H^{k+2} = \{Z\_j-Z\_{k+j+2}=0,\: j = 1,\dots,k-1\}\cong\mathbb{P}^{k+2}$$
then we have 
$$S\_{(1,k)} = \Sigma\_{(1,k)}\cap H^{k+2}\subset\mathbb{P}^{2k+1}$$


