Question

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.

Answer 1

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).