Pluricanonical Geometry of Varieties Isogenous to a Product and Abelian Covers
Massimiliano Alessandro, Davide Frapporti, Christian Gleissner

TL;DR
This paper investigates the properties of pluricanonical maps of varieties isogenous to a product, providing a decomposition theorem for abelian covers and establishing birationality results for threefolds, along with explicit examples and computational classifications.
Contribution
It introduces a decomposition theorem for pluricanonical systems of abelian covers and applies it to study the birationality of pluricanonical maps of varieties isogenous to a product.
Findings
The 4-canonical map of threefolds is birational for p_g ≥ 5.
Constructed examples attain maximal canonical degree with isolated non-normal singularities.
Computational classifications show non-birational bicanonical maps without genus-2 fibrations.
Abstract
We study canonical and pluricanonical maps of varieties isogenous to a product of curves, i.e., quotients of the form with and acting freely. For this purpose, we provide a technical result which is of general interest: a decomposition theorem for pluricanonical systems of abelian covers. This theorem provides an effective tool for the explicit study of geometric properties, such as base loci and the birationality of pluricanonical maps. For threefolds isogenous to a product, we prove that the 4-canonical map is birational for and construct an example attaining the maximal canonical degree for this class of threefolds. In this example, the canonical map is the normalization of its image, which admits isolated non-normal singularities. Computational classifications also reveal threefolds where the bicanonical map fails…
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Geometry and complex manifolds · Advanced Algebra and Geometry
