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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.

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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.

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  • $\begingroup$ "A general cubic $V$" means a general $P^1 \times P^2$, I guess. $\endgroup$
    – Sasha
    Commented Sep 28, 2016 at 15:19
  • $\begingroup$ Yes, right. I have edited. $\endgroup$
    – abx
    Commented Sep 28, 2016 at 15:27

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