Steiner representations of hypersurfaces
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866913560232198144 |
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| author | Antonelli, Vincenzo Casnati, Gianfranco |
| author_facet | Antonelli, Vincenzo Casnati, Gianfranco |
| contents | 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 $φ\colon \mathcal G(-1)\to\mathcal F^\vee$ such that the form defining $X$ to the power $\mathrm{rk}(\mathcal E)$ is exactly $\det(φ)$. 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 $φ\colon \mathcal F(-1)\to\mathcal F^\vee$, where $\mathcal F$ is a suitable Steiner bundle on $\mathbb P^3$ of sufficiently large even rank. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_03408 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Steiner representations of hypersurfaces Antonelli, Vincenzo Casnati, Gianfranco Algebraic Geometry Commutative Algebra Primary: 14F06. Secondary: 14D21, 14J60, 14M12 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 $φ\colon \mathcal G(-1)\to\mathcal F^\vee$ such that the form defining $X$ to the power $\mathrm{rk}(\mathcal E)$ is exactly $\det(φ)$. 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 $φ\colon \mathcal F(-1)\to\mathcal F^\vee$, where $\mathcal F$ is a suitable Steiner bundle on $\mathbb P^3$ of sufficiently large even rank. |
| title | Steiner representations of hypersurfaces |
| topic | Algebraic Geometry Commutative Algebra Primary: 14F06. Secondary: 14D21, 14J60, 14M12 |
| url | https://arxiv.org/abs/2210.03408 |