Deformations of the Fano scheme of a cubic
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866911179727699968 |
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| author | Stark, Samuel |
| author_facet | Stark, Samuel |
| contents | We study the deformation theory of the Fano scheme $\mathrm{F}=\mathrm{F}(\mathrm{X})$ of lines on a cubic $\mathrm{X}$ of dimension $d$ with only finitely many singularities. By taking the relative Fano scheme, we define a morphism $η:\mathscr{D}_{\mathrm{X}}\rightarrow\mathscr{D}_{\mathrm{F}}$ of the local moduli functors associated to $\mathrm{X}$ and $\mathrm{F}$, respectively. We show that for $d\geqslant 5$, $η$ yields an isomorphism on first-order deformations; in particular, $η$ is an isomorphism whenever $\mathrm{H}^{0}(Θ_{\mathrm{X}})=0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_08762 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Deformations of the Fano scheme of a cubic Stark, Samuel Algebraic Geometry We study the deformation theory of the Fano scheme $\mathrm{F}=\mathrm{F}(\mathrm{X})$ of lines on a cubic $\mathrm{X}$ of dimension $d$ with only finitely many singularities. By taking the relative Fano scheme, we define a morphism $η:\mathscr{D}_{\mathrm{X}}\rightarrow\mathscr{D}_{\mathrm{F}}$ of the local moduli functors associated to $\mathrm{X}$ and $\mathrm{F}$, respectively. We show that for $d\geqslant 5$, $η$ yields an isomorphism on first-order deformations; in particular, $η$ is an isomorphism whenever $\mathrm{H}^{0}(Θ_{\mathrm{X}})=0$. |
| title | Deformations of the Fano scheme of a cubic |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2207.08762 |