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This is a Quantitative Study study.

October 9, 2025Open Access

Quartic curves in the quintic del Pezzo threefold

Key Points

The Hilbert scheme of quartic curves on the quintic del Pezzo threefold is smooth and irreducible.
It is shown that the Hilbert scheme is isomorphic to a Grassmannian bundle over another Hilbert scheme.
This work builds upon previous studies of rational quartic curves related to the moduli space of stable maps.
The findings enhance understanding of geometry in the context of rational quartic curves on del Pezzo surfaces.

Abstract

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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