It is known (see for instance [Beauville - Determinantal hypersurfaces](https://projecteuclid.org/euclid.mmj/1030132707)) 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?



Pfaffian representation of the Fermat quintic
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


