Pfaffian representation of the Fermat quintic It is known (see for instance Beauville - Determinantal hypersurfaces) that a generic homogeneous polynomial in $5$ variables of degree $5$ with complex coefficients can be written as the Pfaffian of a skew-symmetric $10 \times 10$ matrix with linear entries in the variables.
I was wondering if such a Pfaffian representation was known to exist for the Fermat quintic plynomial $F_5(X_1,X_2,X_3,X_4,X_5) = X_1^5 + X_2^5 + X_3^5 +X_4^5 + X_5^5$? If so, is there any reference which makes it explicit?
 A: This is not a complete answer, but I will give a concrete computation that shows that the Fermat quintic is in the adherence of the locus of Pfaffian quintics. Of course, this is a trivial consequence of Beauville's result (and Schreyer's computations with Macaulay2). But the proof I give is computer-free and might be of interest for concrete computations in this area. In fact, I will prove something more (which is not known to the best of my knowledge), the Fermat quintic in 6 variables is in the adherence of the Pfaffian locus.
The equivalence between $10 \times 10$ skew symmetric matrices and $5 \times 5$ hermitian matrices with quaternionic coefficients is well-known, so I might just be looking at $5 \times 5$ hermitian matrices with quaternionic coefficients. Let $t$ be a real paramater (which will go to $0$), $\sigma_1, \sigma_2, \sigma_3, \theta_1, \theta_2, \theta_3$ six quaternions and $l_1,l_2,l_3$ three real numbers. Let $M_t$ be the $5 \times 5$ hermitian matrices defined by:
$$M_t = \left(
\begin{array}{ccccc} l_1 & 0 & \sigma_3 & 0 & \sigma_2 \\ 0 & l_2 & \sigma_1 & \theta_3 & 0 \\ \overline{\sigma_3} & \overline{\sigma_1} & 0 & \theta_2& 0 \\ 0 & \overline{\theta_3} & \overline{\theta_2} & 0 & \theta_1 \\ \overline{\sigma_2} & 0 &0& \overline{\theta_1} & l_3\\
\end{array}
\right)
$$
A tedious but simple computations shows that :
$$ det(M_t) = l_1 \sigma_1 \overline{\sigma_1} \theta_1 \overline{\theta_1} + l_2 \sigma_2 \overline{\sigma_2} \theta_2 \overline{\theta_2} + l_3 \sigma_3 \overline{\sigma_3} \theta_3 \overline{\theta_3} + l_1l_3 Re(\sigma_1 \theta_2 \overline{\theta_3}) - l_2Re(\sigma_3 \theta_2 \theta_1 \overline{\sigma_2}) + Re(\sigma_3 \theta_3 \overline{\sigma_1} \theta_1 \overline{\sigma_2}) - \sigma_2 \overline{\sigma_2}Re(\sigma_1 \theta_2 \overline{\theta_3})
+ l_2 \sigma_3 \overline{\sigma_3} \theta_1 \overline{\theta_1} - l_1l_2l_3 \overline{\theta_2} \theta_2.$$
If we let $l_1 = X_1+X_2$, $\sigma_1 = j\times(X_1 + \omega X_2)$, $\theta_1 = X_1 + \omega^2 X_2$, $l_2 = t^2(X_3+X_4)$, $\sigma_2 = \frac{1}{t} (X_3 + \omega X_4)$, $\theta_2 = X_3 + \omega^2 X_4$, $l_3 = X_5+X_6$, $\sigma_3 = X_5 + \omega X_6$, $\theta_3 = X_5 + \omega^2 X_6$, where $\omega = e^{i\frac{\pi}{5}}$ and $j$ is the "second quaternionic square root of $-1$".
Then we get:
$$ det(M_t) = X_1^5 + X_2^5 + X_3^5 + X_4^5 + X_5^5 + X_6^5 - t(X_3+X_4)\mathrm{Re}((X_5+\omega X_6)(X_3+ \omega^2 X_4)(X_1+\omega^2 X_2)(X_3+\overline{\omega}X_4)) + t^2(X_3+X_4)(X_5+\omega X_6)(X_5+\overline{\omega} X_6)(X_1+\omega^2 X_2)(X_1+\overline{\omega^2} X_2) -t^2(X_1+X_2)(X_3+X_4)(X_5+X_6)(X_3+ \omega^2X_4)(X_3+\overline{\omega^2}X_4).$$ 
The limit when $t \longrightarrow 0$ of $det(M_t)$ is $X_1^5+X_2^5+X_3^5+X_4^5+X_5^5+X_6^5$, which shows that the Fermat quintic in 6 variables is in the adherence of the Pfaffian locus.
Remark : The Fermat quintic in $7$ variables can't be in the adherence of the Pfaffian locus. Indeed, any six dimensional linear sections of the Pfaffian quintic in $\mathbb{P}^{44}$ is singular. But the Fermat quintic is always smooth, so it can't be in the adherence of a family of singular qintics. This shows that the "curve" of matrices found above is opitmal in some sense
