Local structure of the Hilbert scheme of conics in quintic del Pezzo varieties

TL;DR

This work proves that the Hilbert scheme of conics on the quintic del Pezzo fourfold $X$ is a smooth, rational variety of dimension $7$, using a $\mathbb{C}^*$-torus action to reduce the problem to fixed conics. It builds a birational model for $\mathbf{H}_2(X)$ via a Grassmannian-bundle construction $\mathbf{S}(X)$ and analyzes the deformation spaces through normal bundles, Ferrand doubling, and BB localization. The authors classify torus-fixed conics (including double and reducible conics) and compute their restricted normal bundles to verify vanishing obstructions, thereby establishing smoothness and rationality. The results reinforce and extend prior indirect proofs, providing a direct deformation-theoretic and birational picture for conics on quintic del Pezzo varieties and offering explicit descriptions of fixed loci and tangent-space weights.

Abstract

Let $X$ be the quintic del Pezzo $4$-fold. It is very well-known that $X$ is realized by a smooth linear section of Grassmannian $\mathrm{Gr}(2,5)$. In this paper, we prove that the Hilbert scheme of conics in $X$ is a smooth variety of dimension $7$ by using a torus action on $X$, which provides a more direct proof about the first named author's previous result.

Theorems & Definitions (34)

• Theorem 1.1: Section 3.4
• Lemma 2.1
• Lemma 2.2: CKK25
• Theorem 2.3: Bir73
• Proposition 2.4: Pro94 and CHL18
• Definition 2.5
• Proposition 2.6
• proof
• Remark 2.7
• Proposition 2.8: CHL18