(Rational normal scroll)
Let \(a_1, \ldots, a_k \geq 1\) and \(N = \sum_i a_i + k - 1\). Choose complementary linear subspaces \(\PP^{a_1}, \ldots, \PP^{a_k} \subseteq \PP^N\), a rational normal curve \(C_i \subseteq \PP^{a_i}\) of degree \(a_i\), and isomorphisms \(\phi_i \colon \PP^1 \to C_i\). The rational normal scroll \(S(a_1, \ldots, a_k) \subseteq \PP^N\) is the union over \(t \in \PP^1\) of the \((k-1)\)-planes spanned by \(\phi_1(t), \ldots, \phi_k(t)\).
\(S(a_1, \ldots, a_k)\) is the image of the projective bundle \(\PP\big(\OO_{\PP^1}(a_1) \oplus \cdots \oplus \OO_{\PP^1}(a_k)\big)\) under the complete linear system of its tautological bundle \(\OO(1)\). It is a smooth variety of dimension \(k\) and degree \(\sum_i a_i\), and its projective equivalence class depends only on the multiset \(\{a_1, \ldots, a_k\}\).
\(S(1,1) \subseteq \PP^3\) is the smooth quadric surface, ruled by the lines joining corresponding points of two skew lines. \(S(1,2) \subseteq \PP^4\) is the cubic scroll, the blowup of \(\PP^2\) at a point embedded by the conics through that point. It is cut out by the \(2 \times 2\) minors of \[ \begin{bmatrix} x_0 & x_1 & y_0 \\ x_1 & x_2 & y_1 \end{bmatrix} , \] and is the closure of the image of \((s, t) \mapsto [1 : s : s^2 : t : st]\). Allowing \(a_i = 0\) gives cones: \(S(0,2)\) is the quadric cone over a conic.
(Scroll)
A scroll is a projective variety \(X \subseteq \PP^N\) with a morphism \(X \to B\) onto a variety \(B\) whose fibres are linear subspaces of \(\PP^N\) of a fixed dimension.