Ciliberto–Di Gennaro conjecture for nodal hypersurfaces

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Let X⊂P⁡4X\subset\operatorname{\mathbb{P}}^4 be a nodal hypersurface, meaning a hypersurface of degree dd whose singularities are all ordinary double points. Assume that XX has at most 2(d−2)(d−1)2(d-2)(d-1) singular points. Ciliberto–Di Gennaro conjecture. One of the following holds: (1) XX is factorial; (2) XX contains a plane and has at least (d−1)2(d-1)^2 nodes; or (3) XX contains a quadric surface and has exactly 2(d−2)(d−1)2(d-2)(d-1) nodes. The conjecture concerns the factoriality of nodal hypersurfaces, a property equivalent here to Q⁡\operatorname{\mathbb{Q}}-factoriality and relevant to questions of rationality. Its status is not resolved in the supplied source context.

References

Primary source

Ksenia Kvitko, “Ciliberto-Di Gennaro conjecture for sextic hypersurfaces”, arXiv:2505.06742 (2025).

Additional references

2 papers in this index state this conjecture (2013–2025). The statement above is taken from the most recent of them; the others are arXiv:1310.0227.

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