Cubic fourfold

In algebraic geometry, a cubic fourfold is a degree 3 hypersurface of dimension 4 in 5-dimensional projective space. Although some cubic fourfolds are known to be rational, it is known that very general cubic fourfolds are not rational, as proven by Katzarkov, Kontsevich, Pantev and Yu.

Using geometric invariant theory (GIT), Radu Laza constructed a compactification of cubic fourfolds with ADE singularities (including all smooth cubic fourfolds). He further showed that this compactification is isomorphic, via the period map, to Looijenga's compactification of the complement of certain arrangement of hyperplanes in the period space. Yuchen Liu further showed that the GIT compactification is isomorphic to the K-moduli of cubic fourfolds.

References

  • Katzarkov, Ludmil; Kontsevich, Maxim; Pantev, Tony; Yu, Tony Yue (2025). "Birational Invariants from Hodge Structures and Quantum Multiplication". arXiv:2508.05105 [math.AG].
  • Laza, Radu (2009). "The moduli space of cubic fourfolds". J. Algebraic Geom. 18 (3). Amer. Math. Soc.: 511–545. arXiv:0704.3256. doi:10.1090/S1056-3911-08-00506-7. Retrieved 29 September 2025.
  • Laza, Radu (2010). "The moduli space of cubic fourfolds via the period map". Ann. of Math. 172. Amer. Math. Soc.: 673–711. arXiv:0705.0949. doi:10.4007/annals.2010.172.673. Retrieved 29 September 2025.
  • Liu, Yuchen (2022). "K-stability of cubic fourfolds". Journal für die reine und angewandte Mathematik (Crelles Journal). 2022 (786): 55–77. doi:10.1515/crelle-2022-0002. Retrieved 29 September 2025.
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