Moduli of Cubic fourfolds and reducible OADP surfaces

Michele Bolognesi, Zakaria Brahimi, Hanine Awada
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Abstract

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.
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立方四折和可还原 OADP 表面的模量
在本文中,我们探讨了哈塞特除数 $\mathcalC_8$ 与其他除数 $\mathcal C_i$ 的交集,哈塞特除数 $\mathcalC_8$ 参数化了包含平面 $P$ 的光滑立方四折$X$。值得注意的是,我们研究了与 $\mathcal{C}_{12}$ 和 $\mathcal{C}_{20}$ 交集的不可还原成分。这两个分维分别泛函包含光滑立方卷轴的立方体和光滑维罗尼斯曲面。首先,我们找出两个交点的所有不可还原分量,并根据 $P$ 与另一个曲面的交点来描述泛函的几何形状。然后,我们考虑这些分量中立方体的合理性问题,或者通过寻找投影离开 $P$ 所诱导的二次纤维的合理截面,或者通过寻找在 $X$ 内的可还原一显双点曲面的例子。最后,通过一些麦考莱计算,我们给出了每个分量中立方体的明确方程。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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