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Let $X\subset \mathbb{P}^5$ a smooth pfaffian smooth cubic fourfold hypersurface. It is easy to see that such a hypersurface must contain a quartic scroll surface. I wonder about the inverse question. If a cubic fourfold $X$ contains a quartic scroll, is it a pfaffian?

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How do the (possibly naïve) parameter counts compare? – Noam D. Elkies Oct 15 '11 at 21:28
I believe this question is answered in Beauville-Donagi (although I don't have the article at the moment to verify). – Jason Starr Oct 16 '11 at 1:41
Thank you for your comments. @Noam: In fact, given a quartic scroll in $P^5$, I haven't computed the dimension of the space of cubics in the ideal of the scroll... is it what you meant? @Jason: in fact the question came up to me while reading Be-Do. I seem to understand that une implication is easy, but they don't seem to prove the other. Am I wrong? – IMeasy Oct 16 '11 at 14:11
If you look in Hassett's thesis, then he carefully does the parameter counts that Noam is suggesting. It follows that the Pfaffian cubic fourfolds form a dense Zariski open subset of the moduli space of all smooth cubic fourfolds containing a quartic scroll. But Hassett does not seems to discuss whether every smooth cubic fourfold containing a quartic scroll is Pfaffian. – Jason Starr Oct 16 '11 at 14:15
up vote 3 down vote accepted

Part (a) of Proposition 9.2 in Beauville's "Determinantal Hypersurfaces" paper (Michigan Mathematical Journal 48, 2000) says that a cubic fourfold is linear Pfaffian precisely when it contains a quintic del Pezzo surface. One path to settling your question is to determine whether every cubic fourfold $X$ containing a quartic scroll $Q$ also contains a 2-plane $P$ for which $Q \cup P$ is a degeneration of a quintic del Pezzo in $X.$

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Joe Harris used to have an (unpublished) atlas of cubic fourfolds, which surfaces imply the existence of which other surfaces, etc. When I dig it up, I will let you know if this is discussed in his atlas. – Jason Starr Oct 16 '11 at 18:58
Thank you for your help! @Jason: If you happen to take a look at that Atlas, it would be of great help, thank you. – IMeasy Oct 17 '11 at 9:13
OK I have just realized how this goes. You consider the product variety $P^2 \times P^1$ embedded in $P^5$ via the Segre embedding. The image is a cubic 3-fold $Y$. Then the intersection $Y\cap X$ is a degree 9 surface. If $X$ is pfaffian then the DP quintic is in linkage (liason) with the quartic RNScroll inside this deg 9 surface. That's why all pfaffian cubics are contained in the special cubic divisor $C_{14}$ defined by hassett. $C_{14}$ is exactly the set of cubics containing one 4tic RNScroll. – IMeasy Jun 10 '13 at 13:44

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