Picard bundles and Brill-Noether loci in the compactified Jacobian of a nodal curve (Q2929648)

The Brill-Noether loci parameterizing line bundles of degree \(d\) on a smooth curve that admit at least \(r\) independent sections have been studied intensively in classical algebraic geometry, see [\textit{E. Arbarello} et al., Geometry of algebraic curves. Volume I. New York etc.: Springer-Verlag (1985; Zbl 0559.14017)] for an overview and references. The paper under review studies the Brill-Noether loci on nodal curves.NEWLINENEWLINELet \(X_k\) denote an irreducible projective nodal curve with \(k\) nodes and of genus \(g(X_k)\), and \(X_k^{o}\subset X_k\) the locus of smooth points. Let \(f'_d\) be the Abel-Jacobi map from the \(d\)th product of \(X_k^{o}\) to the compactified Picard variety \(\bar{J}^d(X_k)\) of \(X_k\). For \(0 \leq i \leq g(X_k) - 1\), define the cycles \(W_i\) to be the closure of the image of \(f'_{g(X_k)-i}\). Fixing \(t \in X_k^o\), further identify \(W_i\) with its isomorphic image in \(\bar{J}^0(X_k)\) under translation by \(\mathcal O_{X_k} (-(g(X_k) - i)t)\). The authors first prove that NEWLINE\[NEWLINEW_i = \frac{W_1^i}{i!}NEWLINE\]NEWLINE as cycles modulo numerical equivalence (Theorem 1.1), generalizing the Poincaré formula from smooth curves to nodal curves.NEWLINENEWLINESecondly, consider the Brill-Noether scheme \(B_{X_k}(1,d,r)\) in \(\bar{J}^0(X_k)\) whose underlying set is the set of torsion-free sheaves of rank \(1\) and degree \(d\) on \(X_k\) with at least \(r\) independent sections. The expected dimension of \(B_{X_k} (1, d, r)\) is given by the Brill-Noether number \(\beta_{X_k} (1, d, r) = g(X_k) - r(r - d -1 + g(X_k))\). The authors show that if \(\beta_{X_k} (1, d, r) \geq 0\), then \(B_{X_k} (1, d, r)\) is nonempty (Theorem 1.2).NEWLINENEWLINEFinally, the authors analyze stability of the Picard bundle and connectedness of \(B_{X_k}(1,d,r)\). More precisely, they prove that for \(d\geq 2g(X_k)\), the Picard bundle on \(\bar{J}^0(X_k)\) is stable but not ample, while for the pullback of the Picard bundle to the desingularization of \(\bar{J}^0(X_k)\), the restriction to a general complete intersection subvariety of codimension \(k\) is ample (Theorems 1.4 and 1.5). They use this to show that \(B_{X_k}(1,d,r)\) is connected if \(\beta_{X_k} (1, d, r) > k\). For \(d = 2g(X_k) - 1\), they prove that the Picard bundle is semistable (Theorem 1.6).