The fiber-full scheme
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866909676086493184 |
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| author | Cid-Ruiz, Yairon Ramkumar, Ritvik |
| author_facet | Cid-Ruiz, Yairon Ramkumar, Ritvik |
| contents | We introduce the fiber-full scheme which can be seen as the parameter space that generalizes the Hilbert and Quot schemes by controlling the entire cohomological data. The fiber-full scheme $\text{Fib}_{\mathcal{F}/X/S}^\mathbf{h}$ is a fine moduli space parametrizing all quotients $\mathcal{G}$ of a fixed coherent sheaf $\mathcal{F}$ on a projective morphism $f:X \subset \mathbb{P}_S^r \rightarrow S$ such that $R^i{{f}_*}\left(\mathcal{G}(ν)\right)$ is a locally free $\mathcal{O}_S$-module of rank equal to $h_i(ν)$, where $\mathbf{h} = (h_0,\ldots,h_r) : \mathbb{Z}^{r+1} \rightarrow \mathbb{N}^{r+1}$ is a fixed tuple of functions. In other words, the fiber-full scheme controls the dimension of all cohomologies of all possible twistings, instead of just the Hilbert polynomial. We show that the fiber-full scheme is a quasi-projective $S$-scheme and a locally closed subscheme of its corresponding Quot scheme. In the context of applications, we demonstrate that the fiber-full scheme provides the natural parameter space for arithmetically Cohen-Macaulay and arithmetically Gorenstein schemes with fixed cohomological data, and for square-free Gröbner degenerations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2108_13986 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The fiber-full scheme Cid-Ruiz, Yairon Ramkumar, Ritvik Algebraic Geometry Commutative Algebra 14C05, 14D22, 13D02, 13D07, 13D45 We introduce the fiber-full scheme which can be seen as the parameter space that generalizes the Hilbert and Quot schemes by controlling the entire cohomological data. The fiber-full scheme $\text{Fib}_{\mathcal{F}/X/S}^\mathbf{h}$ is a fine moduli space parametrizing all quotients $\mathcal{G}$ of a fixed coherent sheaf $\mathcal{F}$ on a projective morphism $f:X \subset \mathbb{P}_S^r \rightarrow S$ such that $R^i{{f}_*}\left(\mathcal{G}(ν)\right)$ is a locally free $\mathcal{O}_S$-module of rank equal to $h_i(ν)$, where $\mathbf{h} = (h_0,\ldots,h_r) : \mathbb{Z}^{r+1} \rightarrow \mathbb{N}^{r+1}$ is a fixed tuple of functions. In other words, the fiber-full scheme controls the dimension of all cohomologies of all possible twistings, instead of just the Hilbert polynomial. We show that the fiber-full scheme is a quasi-projective $S$-scheme and a locally closed subscheme of its corresponding Quot scheme. In the context of applications, we demonstrate that the fiber-full scheme provides the natural parameter space for arithmetically Cohen-Macaulay and arithmetically Gorenstein schemes with fixed cohomological data, and for square-free Gröbner degenerations. |
| title | The fiber-full scheme |
| topic | Algebraic Geometry Commutative Algebra 14C05, 14D22, 13D02, 13D07, 13D45 |
| url | https://arxiv.org/abs/2108.13986 |