Moduli of oriented orthogonal sheaves on a nodal curve
type: Item
instance of: scholarly article
title: Moduli of oriented orthogonal sheaves on a nodal curve
zbMATH Open document ID: 1270.14015
DOI: 10.1215/21562261-1966071
author: Takeshi Abe
published in: Kyoto Journal of Mathematics
publication date: 2013-05-10T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.kjm/1364218041
review text: The paper under review is devoted to the study of moduli of oriented orthogonal sheaves on a nodal curve. One of the motivations for this paper comes from the so-called strange duality phenomena on the spaces of generalized theta functions on different moduli spaces. Some typical strange dualities appears in those papers [\textit{A. Beauville}, Ann. Inst. Fourier 56, No. 5, 1405--1418 (2006; Zbl 1114.14021); \textit{A. Marian} and \textit{D. Oprea}, Invent. Math. 168, No. 2, 225--247 (2007; Zbl 1117.14035); \textit{P. Belkale}, J. Am. Math. Soc. 21, No. 1, 235--258 (2008; Zbl 1132.14028)]. Moreover, the author has proved the strange duality for symplectic bundles in [Int. Math. Res. Not. Article ID rnn121, 47 pp (2008; Zbl 1175.14026); J. Reine Angew. Math. 631, 181--220 (2009; Zbl 1177.14068)]. Suppose that a smooth projective curve \(C\) degenerates to an irreducible nodal curve \(C_0\), the factorization theorem says that the space of generalized theta functions on the moduli space of certain sheaves on \(C_0\) decomposes as a direct sum of spaces of generalized theta functions on the moduli spaces of certain parabolic sheaves on the normalization \(\widetilde{C}_0\) of \(C_0\). For precise statements, see [\textit{M. S. Narasimhan} and \textit{T. R. Ramadas}, Invent. Math. 114, No. 3, 565--623 (1993; Zbl 0815.14014)] and [\textit{X. Sun}, J. Algebr. Geom. 9, No. 3, 459--527 (1993; Zbl 0971.14030)]. The factorization theorem plays an important rule in the proof of strange duality for symplectic bundles. In order to obtain a similar duality for orthogonal sheaves, the author describes how the space of global sections of a power of the determinant line bundle on the moduli stack of oriented orthogonal sheaves on \(C_0\) decomposes as a direct sum of spaces of global sections of a line bundle on the moduli stack of parabolic oriented orthogonal bundles on \(\widetilde{C}_0\), which is the main result of this paper (the decomposition involves so many definitions in t…
Mathematics Subject Classification ID: 14H60
Mathematics Subject Classification ID: 14D20
zbMATH DE Number: 6162103
zbMATH Keywords: orthogonal sheaves
zbMATH Keywords: moduli stack
zbMATH Keywords: factorization theorem
reviewed by: Lingguang Li
MaRDI profile type: Publication
OpenAlex ID: W1974977534
cites work: Q3568173
cites work: Strange Duality for Parabolic Symplectic Bundles on a Pointed Projective Line
cites work: Degeneration of the strange duality map for symplectic bundles
cites work: Compactifying the Picard scheme
cites work: Orthogonal bundles on curves and theta functions
cites work: Q4852332
cites work: The strange duality conjecture for generic curves
cites work: Moduli-stacks for bundles on semistable curves
cites work: A modular compactification of the general linear group
cites work: A canonical decomposition of generalized theta functions on the moduli stack of Gieseker vector bundles
cites work: The line bundles on the moduli of parabolic G-bundles over curves and their sections
cites work: The level-rank duality for non-abelian theta functions
cites work: Factorisation of generalised theta functions. I
cites work: Functors of Artin Rings
cites work: Q4498203
cites work: Q3472274
description: scientific article; zbMATH DE number 6162103