Clemens's conjecture for rational curves on the generic quintic threefold

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Let X⊂P4X\subset\mathbb{P}^4 be a generic quintic threefold. A rational curve means a curve on XX birational to P1\mathbb{P}^1, and its degree is measured with respect to the embedding X⊂P4X\subset\mathbb{P}^4. Clemens's conjecture. The generic quintic threefold XX admits only a finite number of rational curves of each degree. Each rational curve is a smoothly embedded P1\mathbb{P}^1 with normal bundle

O(−1)⊕O(−1).\mathscr{O}(-1)\oplus\mathscr{O}(-1).

Furthermore, all the rational curves on XX are mutually disjoint. This is the classical Clemens conjecture for the generic quintic threefold; the supplied passage gives no resolution status for the full statement.

References

Primary source

Rodolfo Aguilar, “The relative Clemens Conjectures for 12-log Calabi-Yau threefolds”, arXiv:2601.11813 (2026).

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