Steiner representations of hypersurfaces

Let $X\subseteq\pmathbb{P}^{n+1}$ be an integral hypersurface of degree $d$. We show that each locally Cohen--Macaulay instanton sheaf $\cE$ on $X$ with respect to $\cO_X\otimes\mathcal{O}_{\mathbb{P}^{n+1}}(1)$ in the sense of [V. Antonelli and G. Casnati, Instanton sheaves on projective schemes, J. Pure Appl. Algebra 227 (2023) 107246, Definition 1.3] yields the existence of Steiner bundles $\mathcal{G}$ and $\mathcal{F}$ on $\mathbb{P}^{n+1}$ of the same rank $r$ and a morphism $\varphi\colon \cG(-1)\to\cF^\vee$ such that the form defining $X$ to the power $\rk(\mathcal{E})$ is exactly $\det(\varphi)$. We inspect several examples for low values of $d$, $n$ and $\rk(\mathcal{E})$. In particular, we show that the form defining a smooth integral surface in $\mathbb{P}^3$ is the pfaffian of some skew--symmetric morphism $\varphi\colon \cmathcal{F}(-1)\to\mathcal{F}^\vee$, where $\mathcal{F}$ is a suitable Steiner bundle on $\mathbb{P}^3$ of sufficiently large even rank.