MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
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 $X.$

share|cite|improve this answer
Joe Harris used to have an (unpublished) atlas of cubic fourfolds, which surfaces imply the existence of which other surfaces, etc.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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.