Conditions for the existence of semistable sheaves on primitive multiple curves (Q627397)

Let \(Y\) be a primitive multiple curve of multiplicity \(n\geq2\) with smooth projective reduced associated curve \(C\) (over \(\mathbb{C}\)) of genus \(g\geq2\). This article provides sufficient criteria for the (semi)stability of torsion free coherent sheaves \(\mathcal{E}\) over \(Y\) in terms of (semi)stability of the restricted sheaves \(\mathcal{E}|_{C}\). In particular, the author shows that, if \(E\) is a vector bundle over \(Y\), and \(E|_{C}\) is semistable (resp. stable), then \(E\) is also semistable (resp. stable). Denote by \(\mathcal{M}(R,d)\) the moduli space of stable torsion free sheaves (not necessarily irreducible) with generalized rank \(R\) and generalized degree \(d\) over \(Y\) (see the construction in [\textit{J.-M. Drézet}, Math. Nachr. 282, No. 7, 919--952 (2009; Zbl 1171.14010)]. These stability criteria are then used to provide conditions on \(n\), \(R\) and \(d\), such that \(\mathcal{M}(R,d)\) is non-empty, by proving the existence of stable quasi-locally free sheaves with certain discrete invariants.