# Real Pfaffian representations of real cubic surfaces

Consider the following classical construction (which is called Pfaffian representation as Sasha indicates):

Let $V^4\subset \Lambda^2 \mathbb R^6$ be a four-dimensional subspace of the space of alternating two-forms. Then the equation $a\wedge a\wedge a=0$, $a\in V^4$ is homogeneous of degree three and hence defines a cubic surface in ${\mathbb P}V^4$.

Question. Can every real cubic surface be obtained by the above construction? If yes, is the set of corresponding representations for each cubic connected? I would be grateful for a reference if there is one.

I am primarily interested in real case but if you only can comment on the complex case, this would be interesting for me as well.

As Sasha says, such cubics are called Pfaffian cubics.

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## 1 Answer

In other words you are asking about Pfaffian representations of a cubic hypersurface. To find a Pfaffian representation of a smooth cubic hypersurface of dimension $d$ is equivalent to constructing a vector bundle $E$ of rank 2 with $c_1 = 2$ generated by $6$ global sections and with $H^\bullet(S,E(-k)) = 0$ for $1 \le k \le d$. So, you are asking whether the moduli space of such bundles on a cubic surface is nonempty and connected.

As far as I know the moduli space is nonempty even for 3-dimensional cubics (and a fortiori for cubic surfaces). Also I know that in dimension 3 the moduli space is connected (in fact it is birational to the intermediate Jacobian of the cubic 3-fold).

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Sasha thank you for the answer in the complex case. I still wonder if this space will be always non-empty in real case. –  aglearner Dec 17 '12 at 14:15
Myself I don't know, but may be you can find out something by looking at arxiv.org/abs/1203.0999. –  Sasha Dec 18 '12 at 3:42
Sasha, thanks again! I will look into this reference and also to the article of Beauville that it cites. –  aglearner Dec 18 '12 at 7:45