Hilbert Scheme of a Pair of Skew Lines on a Cubic Hypersurface
| Autor: | Zhang, Yilong |
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| Rok vydání: | 2025 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | We study an irreducible component $H(X)$ of the Hilbert scheme $Hilb^{2t+2}(X)$ of a smooth cubic hypersurface $X$ containing two disjoint lines. For cubic threefolds, $H(X)$ is always smooth, as shown in arXiv:2010.11622. We provide a second proof and generalize this result to higher dimensions. Specifically, for cubic hypersurfaces of dimension at least four, we show $H(X)$ is normal, and it is smooth if and only if $X$ lacks certain "higher triple lines." Using Hilbert-Chow morphism, we describe the birational geometry of $H(X)$. As an application, when $X$ is a general cubic fourfold, a natural double cover of $H(X)$ yields a smooth model extending a rational map considered by Voisin. Comment: 30 pages, 2 figures, comments welcome |
| Databáze: | arXiv |
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