Derived categories of curves of genus one and torsors over abelian varieties

2022/12/30 by Niranjan Ramachandran, Ramachandran, Niranjan, Rosenberg, Jonathan
Mathematics · Physics and Astronomy · #14F08 (Primary) 14H52 #14F22 #18G80 #81T30 #81T35 (Secondary) #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2212.14497

openalex publication_date 2022/12/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Suppose C is a smooth projective curve of genus 1 over a perfect field F, and E is its Jacobian. In the case that C has no F-rational points, so that C and E are not isomorphic, C is an E-torsor with a class δ(C)∈ H¹(Gal( F/F), E( F)). Then δ(C) determines a class β∈ Br(E)/Br(F) and there is a Fourier-Mukai equivalence of derived categories of (twisted) coherent sheaves \mathcal D(C) \xrightarrow≅ \mathcal D(E, β^-1). We generalize this result to higher dimensions; namely, we prove it also for torsors over abelian varieties.

Related

• Derived categories of torsors for abelian schemes
• Derived categories of irreducible projective curves of arithmetic genus one
• p-torsion étale sheaves on the Jacobian of a curve
• Semi-Homogeneous Sheaves and Twisted Derived Categories
• Autoequivalences of derived categories of elliptic surfaces with non-zero Kodaira dimension