On the Poincaré formula and the Riemann singularity theorem over nodal curves
type: Item
instance of: scholarly article
title: On the Poincaré formula and the Riemann singularity theorem over nodal curves
zbMATH Open document ID: 1184.14040
author: Usha N. Bhosle
author: A. J. Parameswaran
published in: Mathematische Annalen
publication date: 2008-12-01T00:00:00Z
review text: Let \(W_d\) denote the cycle on the Jacobian of a smooth projective curve of genus \(g\) given by the image of the \(d\)th symmetric product of the curve under the Abel-Jacobi map. Poincaré's formula says that \(W_{g-d}={W^d_{g-1}\over d!}\). The present paper gives an analogue of this formula for a nodal curve \(X\) of arithmetic genus \(g\) with \(k\) nodes in characteristic 0. Let \(\overline J^0(X)\) denote the moduli space of rank-1 torsion free sheaves of degree 0 on \(X\) and \(\widetilde J^0(X)\) its normalization. Using the symmetric powers of the smooth part of \(X\) the autors define cycles \(\widetilde W_d\) in \(\widetilde J^0(X)\) and proved the analogue of Poincaré's formula \(\widetilde W_{g-d}= {\widetilde W^d_{g-1}\over d!}\). Moreover, identifying \(\widetilde J^0(X)\) with the corresponding Brill-Noether locus \(\widetilde B_X(1,d,1)\), also an analogue of the Riemann singularity theorem is shown, namely: (1) \(\widetilde B_X(1,d,1)\) is a normal projective variety for all \(d\), (2) \(\widetilde B_X(1,d,2)\) is the singular locus of \(\widetilde B_X(1,d,1)\) for all \(d< g\) and (3) for \(d< g\), \(r\geq 2\), \(\widetilde B_X(1,d,r)\) has codimension \(\geq r\) in \(\widetilde B_X(1,d,1)\). The proofs use some results on compactifications of the Picard scheme as well as on parabolic bundles of the first author.
reviewed by: Herbert Lange
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 14H40
Mathematics Subject Classification ID: 14H20
zbMATH DE Number: 5375984
zbMATH Keywords: nodal curve
zbMATH Keywords: Poincaré's theorem
zbMATH Keywords: Riemann's singularity theorem
MaRDI profile type: Publication
full work available at URL: s00208-008-0259-7
OpenAlex ID: W2064189456
cites work: Q3347989
cites work: Compactifying the Picard scheme
cites work: Generalised parabolic bundles and applications to torsionfree sheaves on nodal curves
cites work: On the Existence of Special Divisors
cites work: On Symmetric Products of Curves
cites work: Symmetric Products and Jacobians
cites work: Compactifications of the Generalized Jacobian Variety
DOI: 10.1007/S00208-008-0259-7
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description: scientific article; zbMATH DE number 5375984