I'm trying to understand why every Pfaffian cubic fourfold contains a rational normal quartic scroll.


I believe this is a well-known classical construction (for example, see Hassett's paper on "special cubic fourfolds"), but I haven't been able to find a reference explaining explicitly how to construct the quartic scroll in a given Pfaffian cubic. Please let me know if you are familiar with this, or if you know of a good reference.



Why do Pfaffian cubic fourfolds contain quartic scrolls?
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](https://projecteuclid.org/download/pdf_1/euclid.mmj/1030132707), § 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. 


