Koszul property of Ulrich bundles and rationality of moduli spaces of stable bundles on Del Pezzo surfaces
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| Formato: | Preprint |
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2022
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| _version_ | 1866916090204913664 |
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| author | Bangere, Purnaprajna Mukherjee, Jayan Raychaudhury, Debaditya |
| author_facet | Bangere, Purnaprajna Mukherjee, Jayan Raychaudhury, Debaditya |
| contents | Let $\mathcal{E}$ be a vector bundle on a smooth projective variety $X\subseteq\mathbb{P}^N$ that is Ulrich with respect to the hyperplane section $H$. In this article, we study the Koszul property of $\mathcal{E}$, the slope-semistability of the $k$-th iterated syzygy bundle $\mathcal{S}_k(\mathcal{E})$ for all $k\geq 0$ and rationality of moduli spaces of slope-stable bundles on Del Pezzo surfaces. As a consequence of our study, we show that if $X$ is a Del Pezzo surface of degree $d\geq 4$, then any Ulrich bundle $\mathcal{E}$ satisfies the Koszul property and is slope-semistable. We also show that, for infinitely many Chern characters ${\bf v}=(r,c_1, c_2)$, the corresponding moduli spaces of slope-stable bundles $\mathfrak{M}_H({\bf v})$ when non-empty, are rational, and thereby produce new evidences for a conjecture of Costa and Miró-Roig. As a consequence, we show that the iterated syzygy bundles of Ulrich bundles are dense in these moduli spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_13631 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Koszul property of Ulrich bundles and rationality of moduli spaces of stable bundles on Del Pezzo surfaces Bangere, Purnaprajna Mukherjee, Jayan Raychaudhury, Debaditya Algebraic Geometry 14J60, 14J26 Let $\mathcal{E}$ be a vector bundle on a smooth projective variety $X\subseteq\mathbb{P}^N$ that is Ulrich with respect to the hyperplane section $H$. In this article, we study the Koszul property of $\mathcal{E}$, the slope-semistability of the $k$-th iterated syzygy bundle $\mathcal{S}_k(\mathcal{E})$ for all $k\geq 0$ and rationality of moduli spaces of slope-stable bundles on Del Pezzo surfaces. As a consequence of our study, we show that if $X$ is a Del Pezzo surface of degree $d\geq 4$, then any Ulrich bundle $\mathcal{E}$ satisfies the Koszul property and is slope-semistable. We also show that, for infinitely many Chern characters ${\bf v}=(r,c_1, c_2)$, the corresponding moduli spaces of slope-stable bundles $\mathfrak{M}_H({\bf v})$ when non-empty, are rational, and thereby produce new evidences for a conjecture of Costa and Miró-Roig. As a consequence, we show that the iterated syzygy bundles of Ulrich bundles are dense in these moduli spaces. |
| title | Koszul property of Ulrich bundles and rationality of moduli spaces of stable bundles on Del Pezzo surfaces |
| topic | Algebraic Geometry 14J60, 14J26 |
| url | https://arxiv.org/abs/2202.13631 |