Constructing prime $\mathbb{Q}$-Fano threefolds of codimension four via key varieties related with $\mathbb{P}^2\times \mathbb{P}^2$-fibrations
Hiromichi Takagi
TL;DR
The paper develops a construction of prime $\mathbb{Q}$-Fano $3$-folds of codimension four by using weighted complete intersections inside weighted projectivizations of two key affine varieties, $\Sigma_{\mathbb{A}}^{13}$ and $\Pi_{\mathbb{A}}^{14}$. It proves that for 23 (resp. 8) numerical data in GRDB there exist quasi-smooth $3$-folds $X$ with $-K_X=\mathcal{O}_X(1)$ and that a general member $T\in|-K_X|$ is a quasi-smooth $K3$ surface with only Du Val singularities; moreover, $\mathrm{Sing}\,T=\mathrm{Sing}\,X$ and $X$ sits inside a larger fibred structure linked to $\mathbb{P}^2\times\mathbb{P}^2$ fibrations. The strategy combines a boundary-based reduction, Bertini-type arguments, and Linear Part Computations (LPC) to control singularities, supported by explicit computer-assisted verifications (Magma/Mathematica). The results populate 23 and 8 GRDB classes with explicit, new examples, enriching the landscape of prime $\mathbb{Q}$-Fano $3$-folds and connecting to prior unprojection and cluster-variety frameworks. This advances the program of constructing and classifying high-dimensional Fano varieties by geometric and computational means, with clear implications for GRDB data and future unprojection-based generations of examples.
Abstract
In our previous research, we constructed the affine varieties $Σ_{\mathbb{A}}^{13}$ and $Π_{\mathbb{A}}^{14}$ 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 $Σ_{\mathbb{A}}^{13}$ (resp.$Π_{\mathbb{A}}^{14}$ 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.