Pluricanonical Geometry of Varieties Isogenous to a Product and Abelian Covers

2025/12/24 by Massimiliano Alessandro, Alessandro, Massimiliano, Davide Frapporti +3
Mathematics · #Algebraic Geometry and Number Theory #Geometry and complex manifolds #Advanced Algebra and Geometry

paper · doi:10.48550/arxiv.2512.21294

Abstract

We study canonical and pluricanonical maps of varieties isogenous to a product of curves, i.e., quotients of the form X = (C₁ × … × C_n)/G with g(C_i)≥ 2 and G 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 p_g ≥ 5 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 to be birational, even in the absence of genus-2 fibrations. This illustrates an interesting phenomenon similar to the non-standard case for surfaces.

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Related

• Pluricanonical maps for threefolds of general type
• The pluricanonical systems of a product-quotient variety
• On pluricanonical maps for threefolds of general type, II
• Finiteness of projective pluricanonical representation for automorphisms of complex manifolds
• Boundedness of fibers for pluricanonical maps of varieties of general type