[Paper Review] On the Prym map of Galois covering

Abolfazl Mohajer|arXiv (Cornell University)|Apr 20, 2020

Algebraic Geometry and Number Theory8 references4 citations

TL;DR

This paper investigates the Prym map for Galois coverings of algebraic curves with abelian and metabelian Galois groups, establishing conditions under which the differential of the Prym map is injective. Using building data and cohomological techniques, it proves injectivity of the differential for abelian covers when certain line bundle degrees exceed critical thresholds, and extends this to metabelian covers via global generation and surjectivity of multiplication maps on sections.

ABSTRACT

In this paper we consider the Prym variety $P(\widetilde{C}/C)$ associated to a Galois coverings of curves $f:\widetilde{C} o C$ branched at $r$ points. We discuss some properties and equivalent definitions and then consider the Prym map $\mathcal{P}=\mathcal{P}(G,g,r):R(G,g,r) o A_{p,δ}$ with $δ$ the type of the polarization. For Galois coverings whose Galois group is abelian and metabelian (non-abelian) we show that the differential of this map at certain points is injective. We also consider the Abel-Prym map $u:\widetilde{C} o P(\widetilde{C}/C)$ and prove some results for its injectivity. In particular we show that in contrast to the classical and cyclic case, the behavior of this map here is more complicated. The theories of abelian and metabelian Galois coverings play a substantial role in our analysis and have been used extensively throughout the paper.

Motivation & Objective

• To analyze the Prym map for Galois coverings of curves with abelian and metabelian Galois groups.
• To determine conditions under which the differential of the Prym map is injective at specific points in the moduli space.
• To generalize results on the Abel-Prym map from cyclic to non-cyclic Galois covers.
• To use the theory of abelian and metabelian coverings to describe the moduli space R(G,g,r) intrinsically.
• To establish sufficient conditions on line bundles and their global sections ensuring surjectivity of multiplication maps, crucial for injectivity of the differential.

Proposed method

• Utilizes reduced building data for abelian Galois covers to parametrize the moduli space R(G,g,r).
• Applies cohomological criteria via the multiplication map μ: H⁰(ω_C ⊗ L_i^{⌊ni/2⌋}) ⊗ H⁰(ω_C ⊗ L_i^{ni−⌊ni/2⌋}) → H⁰(ω_C²(D_i)) to assess injectivity of the Prym map differential.
• Employs the theory of vector bundles U_χ = q_*F_χ in metabelian covers to generalize the analysis beyond line bundles.
• Relies on [3, Theorem 1] to ensure surjectivity of multiplication maps when line bundles are very ample or globally generated with sufficient degree.
• Uses the condition h⁰(ω_C² ⊗ L'² ⊗ U_χ ⊗ U_χ⁻¹) ≤ t(h⁰(ω_C ⊗ L' ⊗ U_χ) + h⁰(ω_C ⊗ L' ⊗ U_χ⁻¹)) − t² to guarantee surjectivity in the metabelian case.
• Applies results from [4] and [1] on global generation and surjectivity of multiplication maps in the context of vector bundles and line bundles.

Experimental results

Research questions

• RQ1 Under what conditions is the differential of the Prym map injective for abelian Galois covers?
• RQ2 How does the behavior of the Abel-Prym map differ in non-cyclic Galois covers compared to the classical cyclic case?
• RQ3 What conditions on the building data ensure the surjectivity of the multiplication map used in the differential of the Prym map?
• RQ4 Can the injectivity of the Prym map differential be extended from abelian to metabelian Galois coverings?
• RQ5 What role do global sections and degrees of line bundles play in determining the injectivity of the Prym map?

Key findings

• For abelian covers, if d_i ≥ 6 when n_i is even or d_i ≥ 7 when n_i is odd, the differential dP of the Prym map is injective.
• If there exists a character χ such that deg(L_χ) ≥ 3 and deg(L_χ⁻¹) ≥ 3, then dP is injective.
• When L_χ and L_χ⁻¹ are globally generated and deg(L_χ) + deg(L_χ⁻¹) ≥ 5, the differential dP remains injective.
• In the metabelian case, if ω_C ⊗ L' is globally generated and U_χ, U_χ⁻¹ are globally generated with h⁰(ω_C² ⊗ L'² ⊗ U_χ ⊗ U_χ⁻¹) ≤ t(h⁰(ω_C ⊗ L' ⊗ U_χ) + h⁰(ω_C ⊗ L' ⊗ U_χ⁻¹)) − t², then dP is injective.
• The injectivity of dP is established via surjectivity of the multiplication map on global sections, which is guaranteed under the stated degree and global generation conditions.
• The paper provides a counterexample showing that the injectivity of the Abel-Prym map fails in non-cyclic Galois covers when the curve is a g²¹₂ₕ, indicating the necessity of the stated conditions.