Constructing prime $\mathbb{Q}$-Fano threefolds of codimension four via key varieties related with ${\mathbb{P}}^2\times {\mathbb{P}}^2$-fibrations

Abstract

In our previous research, we constructed the affine varieties $\hat{I}{\pounds }_{\mathbb{A}}^{13}$ and ${\hat{I}}_{\mathbb{A}}^{14}$ whose partial projectivizations admit ${\mathbb{P}}^2\times {\mathbb{P}}^2$-fibrations with relative Picard number one. In this paper, we produce prime quasi-smooth $\mathbb{Q}$-Fano 3-folds which are anticanonically embedded of codimension four and belong to 23 (resp.8) classes in the Graded Ring Database [GRDB], as weighted complete intersections in weighted projectivizations of $\hat{I}{\pounds }_{\mathbb{A}}^{13}$ (resp.${\hat{I}}_{\mathbb{A}}^{14}$ or its cone). We also show that a general member of the anticanonical linear system of a general prime $\mathbb{Q}$-Fano $3$-fold constructed in this way is a quasi-smooth $K3$ surface with at worst Du Val singularities.

The contents of this paper consist of those of the part of [Tak2] and [Tak3] where we construct examples of $\mathbb{Q}$-Fano threefolds. arXiv admin note: substantial text overlap with arXiv:2103.11086, arXiv:2111.14328