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 $$S_{(1,k)} = \Sigma_{(1,k)}\cap H^{k+2}\subset\mathbb{P}^{2k+1}$$