Moduli of Cubic fourfolds and reducible OADP surfaces
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arXiv
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| Format: | Preprint |
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2024
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| author | Bolognesi, Michele Brahimi, Zakaria Awada, Hanine |
| author_facet | Bolognesi, Michele Brahimi, Zakaria Awada, Hanine |
| contents | In this paper we explore the intersection of the Hassett divisor $\mathcal C_8$, parametrizing smooth cubic fourfolds $X$ containing a plane $P$ with other divisors $\mathcal C_i$. Notably we study the irreducible components of the intersections with $\mathcal{C}_{12}$ and $\mathcal{C}_{20}$. These two divisors generically parametrize respectively cubics containing a smooth cubic scroll, and a smooth Veronese surface. First, we find all the irreducible components of the two intersections, and describe the geometry of the generic elements in terms of the intersection of $P$ with the other surface. Then we consider the problem of rationality of cubics in these components, either by finding rational sections of the quadric fibration induced by projection off $P$, or by finding examples of reducible one-apparent-double-point surfaces inside $X$. Finally, via some Macaulay computations, we give explicit equations for cubics in each component. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_12032 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Moduli of Cubic fourfolds and reducible OADP surfaces Bolognesi, Michele Brahimi, Zakaria Awada, Hanine Algebraic Geometry In this paper we explore the intersection of the Hassett divisor $\mathcal C_8$, parametrizing smooth cubic fourfolds $X$ containing a plane $P$ with other divisors $\mathcal C_i$. Notably we study the irreducible components of the intersections with $\mathcal{C}_{12}$ and $\mathcal{C}_{20}$. These two divisors generically parametrize respectively cubics containing a smooth cubic scroll, and a smooth Veronese surface. First, we find all the irreducible components of the two intersections, and describe the geometry of the generic elements in terms of the intersection of $P$ with the other surface. Then we consider the problem of rationality of cubics in these components, either by finding rational sections of the quadric fibration induced by projection off $P$, or by finding examples of reducible one-apparent-double-point surfaces inside $X$. Finally, via some Macaulay computations, we give explicit equations for cubics in each component. |
| title | Moduli of Cubic fourfolds and reducible OADP surfaces |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2409.12032 |