This paper proves isomorphism of Hilbert scheme of quartic curves on quintic del Pezzo, suggesting a smooth structure.
In this paper, we prove that the Hilbert scheme H₄(X₅) of rational quartic curves on the quintic del Pezzo threefold X₅ is isomorphic to a Grassmannian bundle over the Hilbert scheme of lines on X₅. In particular, H₄(X₅) is smooth and irreducible. Our approach builds upon the geometry of rational quartic curves on X₅ studied by Fanelli-Gruson-Perrin in their work on the moduli space of stable maps to X₅.
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Chung et al. (2025) studied this question.
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