A MASTER SPACE FOR MODULI SPACES OF GIESEKER-STABLE SHEAVES

2016/05/21 by Daniel Greb, Julius Ross, John Ross +1
Mathematics · Physics and Astronomy · #Algebra over a field #Algebraic Geometry and Number Theory #Geometric and Algebraic Topology #Linguistics #Mathematical analysis #Mathematics #Moduli of algebraic curves #Moduli space #Nonlinear Waves and Solitons #Pure mathematics #Space (punctuation) #math.AG #msc:14D20 #msc:14J60 #msc:14L24 #msc:16G20

paper · pdf · doi:10.1007/s00031-018-9477-6

published as Transformation Groups 24 (2019), no. 2, 379-401 · 18 pages

arxiv created 2016/05/21 · openalex publication_date 2018/01/16 · arxiv updated 2019/06/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We consider a notion of stability for sheaves, which we call multi-Gieseker stability that depends on several ample polarisations L₁, …, L_N and on an additional parameter σ∈ ℚ_{≥ 0}^N∖\0\. The set of semi stable sheaves admits a projective moduli space \mathcal M_σ. We prove that given a finite collection of parameters σ, there exists a sheaf- and representation-theoretically defined master space Y such that each corresponding moduli space is obtained from Y as a Geometric Invariant Theory (GIT) quotient. In particular, any two such spaces are related by a finite number of "Thaddeus-flips". As a corollary, we deduce that any two Gieseker-moduli space of sheaves (with respect to different polarisations L₁ and L₂) are related via a GIT-master space. This confirms an old expectation and generalises results from the surface case to arbitrary dimension.

Citations

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