Classifies irreducible components of Kontsevich moduli spaces for genus one stable maps on degree 4 and 5 del Pezzo threefolds and verifies Geometric Manin's conjecture.
Quartic curves in the quintic del Pezzo threefold
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abstract
In this paper, we prove that the Hilbert scheme $\mathbf{H}_4(X_5)$ of rational quartic curves on the quintic del Pezzo threefold $X_5$ is isomorphic to a Grassmannian bundle over the Hilbert scheme of lines on $X_5$. In particular, $\mathbf{H}_4(X_5)$ is smooth and irreducible. Our approach builds upon the geometry of rational quartic curves on $X_5$ studied by Fanelli-Gruson-Perrin in their work on the moduli space of stable maps to $X_5$.
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Moduli space of genus one curves on quartic and quintic del Pezzo threefolds
Classifies irreducible components of Kontsevich moduli spaces for genus one stable maps on degree 4 and 5 del Pezzo threefolds and verifies Geometric Manin's conjecture.