Steiner representations of hypersurfaces

Let $X\subseteq {\mathbb{P}}^{n+1}$ be an integral hypersurface of degree $d$. The description of hypersurfaces in ${\mathbb{P}}^{n+1}$ as zero loci of suitable square matrices (possibly with some further properties, e.g. with linear entries, symmetric, skew--symmetric, etc.) is a very classical topic in algebraic geometry. In this talk we show that each locally Cohen--Macaulay instanton sheaf $E$ on $X$ with respect to ${\text{cO}}_X\otimes {\text{cO}}_{{\mathbb{P}}^{n+1}}(1)$ yields the existence of Steiner bundles $\text{G}$ and $F$ on ${\mathbb{P}}^{n+1}$ of the same rank $r$ and a morphism $\phi :G(-1){\text{toF}}^{\vee }$ such that the form defining $X$ to the power $\text{rk}(E)$ is exactly $det(\phi )$. In particular, we show that the form defining a smooth integral surface in ${\mathbb{P}}^3$ is the pfaffian of some skew--symmetric morphism $\phi :F(-1)\to F^{\vee }$, where $F$ is a suitable Steiner bundle on ${\mathbb{P}}^3$ of sufficiently large even rank. Finally we deal with the case of cubic fourfolds in ${\mathbb{P}}^5$, showing how the existence of Steiner pfaffian representations is related to the existence of particular subvarieties of the cubic.

This is a joint work with Gianfranco Casnati.