Geography and Deformations of $\mathbb{Z}_2^s$-Covers of General Type Over Weighted Projective Threefolds

Technical Summary: Geography and Deformations of $(\mathbb{Z}_2)^s$-Covers of Weighted Projective Threefolds

Problem Statement
The classification of varieties of general type and the description of their moduli spaces remain central problems in higher-dimensional algebraic geometry. While the theory for surfaces is well-developed, the case of threefolds is less understood, particularly regarding the geography of numerical invariants, the structure of moduli components, and the behavior of pluricanonical maps. This paper investigates threefolds of general type constructed as $(\mathbb{Z}_2)^s$-covers of weighted projective spaces ($WP_3$). The authors address three primary questions:

1. What is the asymptotic behavior of the Chern ratios for such covers?
2. Under what conditions do these covers define open subsets of components in the moduli space of stable varieties?
3. When is the covering map induced by a pluricanonical map (specifically, when is the cover flat and pluricanonical)?

Methodology
The authors build upon the foundational theory of abelian covers developed by Pardini [Par91] and its extension to singular bases by Alexeev and Pardini [VR12]. The methodology combines several distinct technical approaches:

• Asymptotic Analysis of Invariants: The authors calculate the volume ($K_X^3$), holomorphic Euler characteristic ($\chi(\mathcal{O}_X)$), and topological Euler characteristic ($e(X)$) for generic $(\mathbb{Z}_2)^s$-covers. They analyze the ratios $x = e(X)/(24\chi(\mathcal{O}_X))$ and $y = -K_X^3/(24\chi(\mathcal{O}_X))$ as the degrees of the branch divisors tend to infinity.
• Deformation Theory: To study the moduli of these covers, the authors extend deformation criteria to the case where the base $Y$ has isolated singularities and the cover $f: X \to Y$ is non-flat (i.e., the pushforward $f_*\mathcal{O}_X$ splits into reflexive sheaves rather than line bundles). They utilize techniques from Schlessinger [Sch71], Wehler [Weh86], and Ran to establish conditions under which deformations of $X$ retain the structure of a finite cover over a deformation of $Y$.
• Fourier Analysis on Finite Groups: To classify flat pluricanonical covers, the authors introduce techniques from the Fourier transform on the finite group $G = (\mathbb{Z}_2)^s$. They relate the degrees of branch divisors to the degrees of the line bundles in the eigensheaf decomposition via unnormalized Fourier transforms. This allows them to derive combinatorial constraints (moment lemmas) that bound the possible values of $s$ and the pluricanonical index $m$.

Key Contributions and Results

1. Chern Ratios and the "Forbidden Zone"
The authors derive explicit formulas for the asymptotic Chern ratios of $(\mathbb{Z}_2)^s$-covers of weighted projective spaces. They define the curve $SCI(x, y) = y(3x+1) - 4 = 0$, which supports the limits of Chern ratios for smooth complete intersections.

• Result: They prove that the Chern ratios of their constructed covers can lie in the region $SCI(x, y) > 0$.
• Significance: This provides a counterexample to a conjecture by Bruce Hunt [Hun06], who proposed that this region (the "empty zone") contains no smooth threefolds. The authors demonstrate the existence of smooth $(\mathbb{Z}_2)^s$-covers of $\mathbb{P}^3$ (using almost uniform covers) whose invariants lie within this zone.

2. Deformation Theory and Moduli Components
The paper generalizes Wehler's criterion for the smoothness of the forgetful map $\text{Def}(f) \to \text{Def}(X)$ to the case of non-flat covers over singular bases.

• Result: They establish sufficient conditions (involving the vanishing of specific cohomology groups like $H^1(\text{Hom}(\Omega_Y, E))$ and $H^0(\text{Ext}^1(\Omega_Y, E))$) under which any deformation of a $(\mathbb{Z}_2)^s$-cover $X$ remains a $(\mathbb{Z}_2)^s$-cover of a deformation of $Y$.
• Application: Using these criteria, they construct new components of the moduli space of stable threefolds. Specifically, they provide numerical criteria for constructing stable covers branched along hyperplane arrangements in weighted projective spaces, showing that for $s \ge 4$, there exist configurations where the general member of the moduli component is a $(\mathbb{Z}_2)^s$-cover.

3. Classification of Flat Pluricanonical Covers
The authors provide a complete classification of flat $(\mathbb{Z}_2)^s$-covers that are $m$-canonical maps.

• Method: By employing Fourier transform techniques on $G = (\mathbb{Z}_2)^s$, they reduce the classification problem to solving systems of linear equations and analyzing moment constraints.
• Result: They show that for $s \ge 2$, there are exactly 32 deformation types of such covers. The classification is finite for specific pairs of $(s, m)$, including cases where $s=1, 2, 3, 4, 5$ and various $m$.
• Non-Flat Cases: In contrast to the flat case, they demonstrate that non-flat canonical and bicanonical covers exist for arbitrarily large values of $s$. This implies that a complete classification of all such covers is impossible without imposing restrictions on the singularities of the cover.

Significance
The paper makes significant contributions to the geography of threefolds by expanding the known range of Chern ratios and refuting a specific conjecture regarding the "empty zone." It advances the deformation theory of abelian covers by handling non-flat cases over singular bases, thereby enabling the construction of new moduli components for stable threefolds. Finally, the application of Fourier analysis to the classification of pluricanonical maps offers a novel combinatorial framework for understanding the rigidity and flexibility of these geometric structures. The work bridges the gap between the well-understood surface case and the more complex threefold case, providing concrete examples and rigorous criteria for the existence and stability of these varieties.