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 $\cO_X\otimes\cO_{\mathbb{P}^{n+1}}(1)$ yields the existence of Steiner bundles $\G$ and $F$ on $\mathbb{P}^{n+1}$ of the same rank $r$ and a morphism $\varphi\colon G(-1)\toF^\vee$ such that the form defining $X$ to the power $\rk(E)$ is exactly $\det(\varphi)$. In particular, we show that the form defining a smooth integral surface in $\mathbb{P}^3$ is the pfaffian of some skew--symmetric morphism $\varphi\colon F(-1)\to F^\vee$, 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.