Pfaffian representations of cubic surfaces

Let $mathbb{K }$ be a field of characteristic zero. We describe an algorithm which requires a homogeneous polynomial $F$ of degree three in $mathbb{K }[x_{0},x_1,x_{2},x_{3}]$ and a zero ${mathbf{a }}$ of $F$ in $mathbb{P }^{3}_{mathbb{K }}$ and ensures a linear Pfaffian representation of $text{ V}(F)$ with entries in $mathbb{K }[x_{0},x_{1},x_{2},x_{3}]$ , under mild assumptions on $F$ and ${mathbf{a }}$ . We use this result to give an explicit construction of (and to prove the existence of) a linear Pfaffian representation of $text{ V}(F)$ , with entries in $mathbb{K }^{prime }[x_{0},x_{1},x_{2},x_{3}]$ , being $mathbb{K }^{prime }$ an algebraic extension of $mathbb{K }$ of degree at most six. An explicit example of such a construction is given.