Moduli of oriented orthogonal sheaves on a nodal curve
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 the paper; therefore, we will not reproduce it here). The proof is technical and similar to the case of symplectic bundles. However it is more complicated than that. The author also studies the deformation of oriented orthogonal sheaves on a formal neighborhood of a node, and obtains some local structure of the moduli stack of oriented orthogonal sheaves on \(C_0\).
scientific article; zbMATH DE number 6162103
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