The locally free locus of Quot schemes on $\mathbb{P}^1$
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909963143610368 |
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| author | Lin, Feiyang Lysek, Theodore |
| author_facet | Lin, Feiyang Lysek, Theodore |
| contents | We characterize components of the locally free locus $\operatorname{Quot}^{n,d}_{\mathbb{P}^1}(\mathcal{O}(\vec{e}))^{\circ}$ of the Quot scheme associated to any vector bundle on $\mathbb{P}^1$. Specifically, we show that the components are in bijection with certain combinatorial objects which we call strongly stable pairs. Using our explicit understanding of the components, we prove that $\operatorname{Quot}^{n,d}_{\mathbb{P}^1}(\mathcal{O}(\vec{e}))^{\circ}$ is connected, and we give an explicit bound for when $\operatorname{Quot}^{n,d}_{\mathbb{P}^1}(\mathcal{O}(\vec{e}))^{\circ}$ is irreducible. The key ingredient is a combinatorial criterion for when a triple of vector bundles on $\mathbb{P}^1$ arises in a short exact sequence. As a consequence, we prove that in codimension $2$, all integral lattice points in the Boij-Söderberg cone are Betti diagrams of actual modules. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_13386 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The locally free locus of Quot schemes on $\mathbb{P}^1$ Lin, Feiyang Lysek, Theodore Algebraic Geometry Commutative Algebra 14D20, 13D02 We characterize components of the locally free locus $\operatorname{Quot}^{n,d}_{\mathbb{P}^1}(\mathcal{O}(\vec{e}))^{\circ}$ of the Quot scheme associated to any vector bundle on $\mathbb{P}^1$. Specifically, we show that the components are in bijection with certain combinatorial objects which we call strongly stable pairs. Using our explicit understanding of the components, we prove that $\operatorname{Quot}^{n,d}_{\mathbb{P}^1}(\mathcal{O}(\vec{e}))^{\circ}$ is connected, and we give an explicit bound for when $\operatorname{Quot}^{n,d}_{\mathbb{P}^1}(\mathcal{O}(\vec{e}))^{\circ}$ is irreducible. The key ingredient is a combinatorial criterion for when a triple of vector bundles on $\mathbb{P}^1$ arises in a short exact sequence. As a consequence, we prove that in codimension $2$, all integral lattice points in the Boij-Söderberg cone are Betti diagrams of actual modules. |
| title | The locally free locus of Quot schemes on $\mathbb{P}^1$ |
| topic | Algebraic Geometry Commutative Algebra 14D20, 13D02 |
| url | https://arxiv.org/abs/2512.13386 |