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

TL;DR

This paper investigates the properties, invariants, and deformation theory of $Z_2^s$-covers of weighted projective threefolds, providing new criteria, classifications, and counterexamples related to their moduli and canonical maps.

Contribution

It extends deformation criteria to non-flat covers, classifies flat pluricanonical maps using Fourier transforms, and constructs examples with diverse deformation types.

Findings

Derived asymptotic behavior and bounds for invariants.

Counterexample to Bruce Hunt's conjecture about smooth threefolds.

Identified 32 deformation types for $s \\geq 2$ and existence of non-flat covers for large $s$.

Abstract

We study threefolds of general type constructed as $\mathbb{Z}_2^s$-covers of weighted projective spaces with a particular focus on their invariants, deformation theory, and the behavior of the $m$-canonical map. For the invariants, we write the ratios of the volume to the topological and holomorphic Euler characteristics as functions of the ratios of the degree of the branch divisors with respect to the total degree. From this expression, we obtain their asymptotic behavior, bounds, and a counterexample to a conjecture made by Bruce Hunt about the non-existence of smooth threefolds in a forbidden zone. From the perspective of deformation theory, we extend the criterion for such covers to be general in their moduli to the case when the weighted projective threefold has isolated singularities and the cover is non-flat, i.e., the pushforward of the structure sheaf splits as a direct sum of reflexive sheaves as opposed to line bundles. As an application, we present new numerical criteria for constructing components of the moduli spaces of stable threefolds and give concrete examples illustrating their application. Finally, we introduce techniques from Fourier transforms on finite groups to completely classify when a $\mathbb{Z}_2^s$-cover is a flat pluricanonical map. For $s \geq 2$, there are $32$ deformation types. We also show that there exist non-flat canonical and bicanonical $\mathbb{Z}_2^s$-covers for arbitrarily large values of $s$.