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arXiv · 2409.12032

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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Michele Bolognesi, Zakaria Brahimi, Hanine Awada. 2024-09-18. Moduli of Cubic fourfolds and reducible OADP surfaces. https://arxiv.org/abs/2409.12032

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