Steiner representations of hypersurfaces

TL;DR

This paper explores the relationship between hypersurfaces in projective space and Steiner bundles, showing how defining forms can be expressed as determinants or pfaffians of morphisms between Steiner bundles, with concrete examples.

Contribution

It establishes a novel connection between hypersurface defining forms and Steiner bundles via morphisms, extending the understanding of algebraic hypersurfaces.

Findings

Defining forms of hypersurfaces can be expressed as determinants of morphisms between Steiner bundles.

For smooth surfaces in P^3, the defining form is a pfaffian of a skew-symmetric morphism.

Examples for low degree and dimension illustrate the theoretical results.

Abstract

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 $\varphi\colon \mathcal G(-1)\to\mathcal F^\vee$ such that the form defining $X$ to the power $\mathrm{rk}(\mathcal E)$ is exactly $\det(\varphi)$. We inspect several examples for low values of $d$, $n$ and $\mathrm{rk}(\mathcal E)$. In particular, we show that the form defining a smooth integral surface in $\mathbb P^3$ is the pfaffian of some skew-symmetric morphism $\varphi\colon \mathcal F(-1)\to\mathcal F^\vee$, where $\mathcal F$ is a suitable Steiner bundle on $\mathbb P^3$ of sufficiently large even rank.