Secant variety and syzygies of Hilbert scheme of two points
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911682141356032 |
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| author | Yoon, Chiwon Seo, Haesong |
| author_facet | Yoon, Chiwon Seo, Haesong |
| contents | In this paper, we prove that $\mathrm{Sec} (X^{[2]})$ features the identifiability under the Grothendieck-Plücker embedding $X^{[2]} \hookrightarrow \PP^N$ when $X$ is embedded by a $4$-very ample line bundle. We also prove that the embedding $X^{[2]} \hookrightarrow \PP^N$ satisfies Green's condition $(N_p)$ when the embedding of $X$ is positive enough. Accordingly, the singular locus of $\mathrm{Sec} (X^{[2]})$ is exactly $X^{[2]}$ when the embedding of $X$ is positive enough. As an application, we describe the geometry of a resolution of singularities from the secant bundle to $\mathrm{Sec}(X^{[2]})$ when $X$ is a surface. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_07315 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Secant variety and syzygies of Hilbert scheme of two points Yoon, Chiwon Seo, Haesong Algebraic Geometry 14C05, 14N07, 13D02, 14E05 In this paper, we prove that $\mathrm{Sec} (X^{[2]})$ features the identifiability under the Grothendieck-Plücker embedding $X^{[2]} \hookrightarrow \PP^N$ when $X$ is embedded by a $4$-very ample line bundle. We also prove that the embedding $X^{[2]} \hookrightarrow \PP^N$ satisfies Green's condition $(N_p)$ when the embedding of $X$ is positive enough. Accordingly, the singular locus of $\mathrm{Sec} (X^{[2]})$ is exactly $X^{[2]}$ when the embedding of $X$ is positive enough. As an application, we describe the geometry of a resolution of singularities from the secant bundle to $\mathrm{Sec}(X^{[2]})$ when $X$ is a surface. |
| title | Secant variety and syzygies of Hilbert scheme of two points |
| topic | Algebraic Geometry 14C05, 14N07, 13D02, 14E05 |
| url | https://arxiv.org/abs/2403.07315 |