On the Poincaré formula and the Riemann singularity theorem over nodal curves
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.
scientific article; zbMATH DE number 5375984
scientific article