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

TL;DR

This paper proves that the Hilbert scheme of conics in a quintic del Pezzo 4-fold is smooth and 7-dimensional, using torus action for a more direct proof of a known result.

Contribution

It provides a new, more direct proof of the smoothness and dimension of the Hilbert scheme of conics in the quintic del Pezzo 4-fold via torus action.

Findings

Hilbert scheme of conics in X is smooth of dimension 7

Torus action facilitates a direct proof of the scheme's properties

Reinforces previous results with a new proof method

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.