Return to Revisions

The construction I know is somewhat indirect. Let $X\subset \mathbb{P}^5$ be defined by the pfaffian of a skew-symmetic matrix $A$ of linear forms. This gives an exact sequence $$0\rightarrow \mathcal{O}_{\mathbb{P}}(-1)^6\xrightarrow{\ \ A\ \ } \mathcal{O}_{\mathbb{P}}^6\rightarrow E\rightarrow 0$$ where $E$ is a rank 2 vector bundle on $X$. The zero locus of a general section of $E$ is a quintic del Pezzo surface $S$ -- see for instance this paper, § 9. Now take a general 3-dimensional cubic scroll $V_3$ in $\mathbb{P}^5$ containing $S$ (the image of $\mathbb{P}^1\times \mathbb{P}^2$). The residual intersection of $S$ in $X\cap V_3$ is a quartic scroll.