Maximal degree subsheaves of torsion free sheaves on singular projective curves (Q2716153)

Let \(E\) be a torsion-free sheaf of rank \(r\) and degree \(d\) on an integral projective curve with arithmetic genus \(g \geq 2\). Define \(s_k(E) = kd - rd'\), where \(d'\) is the maximum of the degrees of subsheaves of \(E\) of rank \(k\). The main result of the paper is the following generalization of the Nagata-Segre theorem. NEWLINENEWLINENEWLINETheorem 1. If \(E\) is a flat limit of a family of locally free sheaves, then \(s_k(E) \leq gk(r-k)\).NEWLINENEWLINENEWLINEIn particular, if the curve has only planar singularities, then \(s_k(E) \leq k(r-k)g\) for any torsion-free sheaf \(E\). NEWLINENEWLINENEWLINEIn case the curve is smooth, the bound on \(s_k(E)\) was proved by \textit{M. Nagata} [Nagoya Math. J. 37, 191-196 (1970; Zbl 0193.21603)] in rank 2 case and generalised to higher ranks by \textit{S. Mukai} and \textit{F. Sakai} [Manuscr. Math. 52, 251-256 (1985; Zbl 0572.14008)]. The author shows that the bound and related results fail for non-Gorenstein curves, namely: NEWLINENEWLINENEWLINETheorem 2. For every integer \(g \geq 9\), there exists a (non-Gorenstein) integral projective curve \(X\) of arithmetic genus \(g\) and a rank \(2\) torsion-free sheaf on it such that \(s_1(E) >g\). NEWLINENEWLINENEWLINEFor a smooth \(X\) \textit{M. Maruyama} showed that if \(r=2, s_1(E) = g\), then the scheme \(M(E)\) of maximum line subbundles of \(E\) has dimension \(1\) [\textit{M. Maruyama}, ``On classification of ruled surfaces'', Lectures in Mathematics. Tokyo (1970; Zbl 0214.20103)]. The author shows that for every \(g\geq 9\), there exists an integral projective curve and a rank \(2\) torsion-free sheaf \(E\) on it such that \(s_1(E) = g\) and \(\dim M(E) = 0\).