Steiner representations of hypersurfaces

Let $X\subseteq {\mathbb{P}}^{n+1}$ be an integral hypersurface of degree $d$. We show that each locally Cohen-Macaulay instanton sheaf $\mathcal{E}$ on $X$ with respect to ${\mathcal{O}}_X\otimes {\mathcal{O}}_{{\mathbb{P}}^{n+1}}(1)$ in the sense of Definition 1.3 in arXiv:2205.04767 [math.AG] yields the existence of Steiner bundles $\mathcal{G}$ and $\mathcal{F}$ on ${\mathbb{P}}^{n+1}$ of the same rank $r$ and a morphism $\phi \text{\hspace{0.17em}⁣}:\mathcal{G}(-1)\to {\mathcal{F}}^{\vee }$ such that the form defining $X$ to the power $\mathrm{r}\mathrm{k}(\mathcal{E})$ is exactly $\det (\phi )$. We inspect several examples for low values of $d$, $n$ and $\mathrm{r}\mathrm{k}(\mathcal{E})$. In particular, we show that the form defining a smooth integral surface in ${\mathbb{P}}^3$ is the pfaffian of some skew-symmetric morphism $\phi \text{\hspace{0.17em}⁣}:\mathcal{F}(-1)\to {\mathcal{F}}^{\vee }$, where $\mathcal{F}$ is a suitable Steiner bundle on ${\mathbb{P}}^3$ of sufficiently large even rank.