Quartic curves in the quintic del Pezzo threefold

TL;DR

This work identifies the Hilbert scheme $\mathbf{H}_4(X_5)$ of rational quartic curves on the quintic del Pezzo threefold $X_5$ as a $\mathrm{Gr}(3,5)$-bundle over $\mathbf{H}_1(X_5)\cong \mathbb{P}^2$, establishing smoothness and irreducibility. The authors show that every quartic rational curve arises as a residual to a line in a hyperplane section, constructing an injective (indeed isomorphic) map from the Grassmannian bundle to $\mathbf{H}_4(X_5)$ and leveraging moduli-space techniques for stable maps. They develop explicit invariant presentations of $X_5$, classify $\mathbb{C}^*$-fixed low-degree curves to control the geometry, and provide a residual-curve construction that yields a global geometric model for quartics on $X_5$. The results pave the way for applications to Donaldson–Thomas theory on linear sections of Grassmannians and offer a concrete blueprint for similar studies on higher-dimensional Fano varieties.

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 $X_5$.

Figures (2)

• Figure 1: Configuration of $\mathbb{C}^*$-fixed twisted cubic curves
• Figure 2: Reducible rational curves of degree $d\leq 4$

Theorems & Definitions (30)

• Theorem 1.1
• Lemma 2.1
• proof
• Proposition 2.2
• proof
• Lemma 3.1
• proof
• Remark 3.2
• Lemma 4.1: Pro92
• proof