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

TL;DR

This paper constructs new prime $Q$-Fano threefolds of codimension four using key varieties related to $P^2 imes P^2$-fibrations, expanding the classification within the Graded Ring Database and analyzing their associated $K3$ surfaces.

Contribution

It introduces a method to produce prime $Q$-Fano 3-folds as weighted complete intersections in specific key varieties, linking them to known classifications and properties.

Findings

Constructed 23 classes of $Q$-Fano 3-folds in the GRDB.

Produced 8 additional classes via cones and projectivizations.

General members have quasi-smooth $K3$ surfaces with Du Val singularities.

Abstract

In our previous research, we constructed the affine varieties $\Sigma_{\mathbb{A}}^{13}$ and $\Pi_{\mathbb{A}}^{14}$ 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 $\Sigma_{\mathbb{A}}^{13}$ (resp.$\Pi_{\mathbb{A}}^{14}$ 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.