Stability of tautological bundles on symmetric products of curves (Q1996417)

Let \(C\) be a smooth projective curve over \(\mathbb C\) and \(E\) a rank \(r\) vector bundle on it. The subject of this paper is the (semi)stability of the tautological rank \(rn\) bundle \(E^{[n]}\) on the symmetric power \(C^{(n)}\) of \(C\), with respect to the ample class represented by \(C^{(n-1)}+ x \subset C^{n}\). A necessary condition is that \(E\) itself is (semi)stable. So, the question is to see when the converse is true. The case \(E\) a line bundle or \(r\) arbitrary but \(n=2\) was studied in the literature (cf. the references given by the paper under review). The main result of the paper is that \(E^{[n]}\) is stable if \(E\) is stable and the slope \(\mu:=\frac{\mathrm{degree}(E}{r} \not\in [-1, n-1]\) and that \(E^{(n)}\) semistable if \(E\) is semistable and and the slope \(\mu:=\frac{\mathrm{degree}(E}{r} \not\in (-1, n-1)\). The author shows that the conditions on \(\mu\) are sharp.