Moduli of torsionfree sheaves of rank two and odd degree on a nodal hyperelliptic curve
type: Item
instance of: scholarly article
title: Moduli of torsionfree sheaves of rank two and odd degree on a nodal hyperelliptic curve
zbMATH Open document ID: 1268.14032
published in: Beiträge zur Algebra und Geometrie
publication date: 2013-03-19T00:00:00Z
review text: Let \(X\) be an irreducible nodal hyperelliptic curve of arithmetic genus \(g\) defined over an algebraically closed field \(k\) of characteristic \(0\) and let \(L\) be a line bundle of odd degree \(d\) on \(X\). Let \(U_L'(2,d)\) be the moduli space of stable bundles of rank \(2\) and determinant \(L\) on \(X\) and let \(U_L(2,d)\) be the closure of \(U_L'(2,d)\) in the moduli space of torsion-free sheaves of rank \(2\) and degree \(d\). In this paper, the author obtains an explicit geometric description of \(U_L(2,d)\), which generalises that given by \textit{U. V. Desale} and \textit{S. Ramanan} for the case where \(X\) is smooth [Invent. Math. 38, 161--185 (1976; Zbl 0323.14012)]. If \(X\) has \(m\) nodes, it determines a pencil of quadrics in \({\mathbb P}^{2g+1}\) with Segre symbol \([2\cdots 2 1\cdots1\)] with \(2\) repeated \(m\) times and \(1\) repeated \(2g+2-2m\) times. Let \(R\) denote the scheme of \((g-2)\)-dimensional linear subspaces of \({\mathbb P}^{2g+1}\) which are isotropic for every member of the pencil. The scheme \(R\) is Cohen-Macaulay and normal (Proposition 3.7). In the case \(m=1\), let \(R_0\) be the subscheme of \(R\) consisting of those subspaces which contain the unique singular point of the intersection of the pencil of quadrics; then \(R':=R-R_0\) is smooth and irreducible (Corollary 3.6). In the case \(m=1\), the author proves in Theorem 1.2 that \(R\) is the normalisation of \(U_L(2,d)\). In fact there is a morphism \(f:R\to U_L(2,d)\) such that the restrictions of \(f\) to \(R'\) and \(R_0\) are isomorphisms onto \(U_L'(2,d)\) and \(U_L(2,d)-U_L'(2,d)\). For \(g=2\), \(R\) is an intersection of quadrics in \({\mathbb P}^5\) and \(R_0\) is the unique singular point of \(R\). Even in this case, it is not known whether \(U_L(2,d)\) is normal. For general \(m\), Theorem 1.3 states that there is a birational surjective morphism \(f:R\to U_L(n,d)\). In the smooth case, a crucial ingredient in the 1976 proof was the fact that, if \(E\)…
Mathematics Subject Classification ID: 14H60
Mathematics Subject Classification ID: 14D20
zbMATH DE Number: 6146770
zbMATH Keywords: nodal hyperelliptic curves
zbMATH Keywords: vector bundles
zbMATH Keywords: torsion-free sheaves
zbMATH Keywords: moduli spaces
zbMATH Keywords: pencils of quadrics
author: Usha N. Bhosle
reviewed by: P. E. Newstead
MaRDI profile type: Publication
full work available at URL: s13366-012-0107-5
OpenAlex ID: W2008675028
cites work: Q3325847
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cites work: MAXIMAL SUBSHEAVES OF TORSION-FREE SHEAVES ON NODAL CURVES
cites work: Classification of vector bundles of rank 2 on hyperelliptic curves
cites work: Stable bundles of rank 2 and odd degree over a curve of genus 2
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DOI: 10.1007/S13366-012-0107-5
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description: scientific article; zbMATH DE number 6146770