Quartic curves in the quintic del Pezzo threefold

TL;DR

This paper characterizes the Hilbert scheme of rational quartic curves on the quintic del Pezzo threefold as a Grassmannian bundle, establishing its smoothness and irreducibility, based on geometric analysis.

Contribution

It proves an isomorphism between the Hilbert scheme of quartic curves and a Grassmannian bundle over the lines on the threefold, advancing understanding of their geometric structure.

Findings

Hilbert scheme of quartic curves is smooth and irreducible.

Hilbert scheme is isomorphic to a Grassmannian bundle.

Method builds on prior work on rational curves and stable maps.

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 $X_5$.