Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Pluricanonical Geometry of Varieties Isogenous to a Product: Chevalley-Weil Theory and Pluricanonical Decompositions of Abelian Covers

Massimiliano Alessandro, Davide Frapporti, Christian Gleissner

TL;DR

The paper develops a unified framework for canonical and pluricanonical maps of varieties isogenous to a product (VIPs), focusing on unmixed types. It proves a Chevalley–Weil formula for pluricanonical representations and a Pardini-type decomposition for abelian covers, enabling explicit analysis of base loci and birationality. In dimension three, it proves the 4-canonical map is birational when $p_g\ge 5$ and exhibits a threefold whose canonical map is the normalization of its image with a sharp degree bound, alongside extensive computational classifications revealing intricate birational behavior beyond standard fibrations. The work integrates group-theoretic data, representation theory, and explicit constructions to advance understanding of pluricanonical geometry in higher dimensions.

Abstract

We study canonical and pluricanonical maps of varieties isogenous to a product of curves, i.e., quotients of the form $ X = (C_1 \times \dots \times C_n)/G $ with $g(C_i)\ge 2$ and $G$ acting freely. We establish the Chevalley-Weil formula for pluricanonical representations of a curve with a finite group action and a decomposition theorem for pluricanonical systems of abelian covers. These tools allow an explicit study of geometric properties of $X$, 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 $p_g \ge 5$ 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.

Pluricanonical Geometry of Varieties Isogenous to a Product: Chevalley-Weil Theory and Pluricanonical Decompositions of Abelian Covers

TL;DR

The paper develops a unified framework for canonical and pluricanonical maps of varieties isogenous to a product (VIPs), focusing on unmixed types. It proves a Chevalley–Weil formula for pluricanonical representations and a Pardini-type decomposition for abelian covers, enabling explicit analysis of base loci and birationality. In dimension three, it proves the 4-canonical map is birational when and exhibits a threefold whose canonical map is the normalization of its image with a sharp degree bound, alongside extensive computational classifications revealing intricate birational behavior beyond standard fibrations. The work integrates group-theoretic data, representation theory, and explicit constructions to advance understanding of pluricanonical geometry in higher dimensions.

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. We establish the Chevalley-Weil formula for pluricanonical representations of a curve with a finite group action and a decomposition theorem for pluricanonical systems of abelian covers. These tools allow an explicit study of geometric properties of , 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 to be birational, even in the absence of genus-2 fibrations. This illustrates an interesting phenomenon similar to the non-standard case for surfaces.
Paper Structure (13 sections, 23 theorems, 152 equations)

This paper contains 13 sections, 23 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Varieties Isogenous to a Product of Unmixed Type
  3. Group-Theoretic Description of an Unmixed VIP
  4. Pluricanonical Representations of an Unmixed VIP
  5. The Chevalley-Weil Formula for Pluricanonical Representations
  6. Pluricanonical Systems of an Abelian Cover
  7. Pluricanonical Systems of an Unmixed VIP with Abelian Group
  8. Pluricanonical maps of a VIP
  9. Generalities
  10. Representation-Theoretic Methods
  11. Examples and Computational Results
  12. Example of a 3-Fold with Birational Canonical Map
  13. Further Computational Results

Key Result

Theorem A

Let $\chi_{\varphi_m}$ be the character of the $m$-canonical representation and $\chi \in \mathop{\mathrm{Irr}}\nolimits(G)$ be the character of an irreducible representation $\varrho \colon G \to \mathrm{GL}(V)$. If $m\geq 2$, then it holds Here, $n_1, \dotso, n_r$ are the branching indices of the cover $C\to C/G$, the integer $[b]_{n_i} \in \lbrace 0, \ldots, n_i-1\rbrace$ is the congruence cl

Theorems & Definitions (56)

  • Theorem A: Theorem \ref{['Chev-Weil-Pluri']}
  • Theorem B: Theorem \ref{['prop-push-decomp']}
  • Theorem C: Theorem \ref{['4-canonical-map']}
  • Theorem D: Theorem \ref{['3-fold-normalization-map']}
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Theorem 2.4: Riemann's existence theorem
  • Definition 2.5
  • Remark 2.6
  • ...and 46 more