The classification of semistable plane sheaves supported on sextic curves (Q394145)

Let \(\mathrm{M}_{\mathbb{P}^2}(r,\chi)\) be the moduli space of Gieseker semistable sheaves on \(\mathbb{P}^2(\mathbb{C})\) with Hilbert polynomial \(P(m)=rm+\chi\).NEWLINENEWLINEIn [Rev. Roum. Math. Pures Appl. 38, No. 7-8, 635--678 (1993; Zbl 0815.14029)] \textit{J. Le Potier} proved that \(\mathrm{M}_{\mathbb{P}^2}(r,\chi)\) is an irreducible projective variety of dimension \(r^2+1\), smooth at points corresponding to stable sheaves. In [\textit{J.-M. Drézet} and \textit{M. Maican}, Geom. Dedicata 152, 17--49 (2011; Zbl 1236.14012)] and [\textit{M. Maican}, Ill. J. Math. 55, No. 4, 1467--1532 (2011; Zbl 1273.14027)] the spaces \(\mathrm{M}_{\mathbb{P}^2}(4,\chi)\) and \(\mathrm{M}_{\mathbb{P}^2}(5,\chi)\), respectively, have been completely described.NEWLINENEWLINEThe paper under review continues the analysis of these moduli spaces, focusing on the case \(\mathrm{M}_{\mathbb{P}^2}(6,\chi)\) corresponding to sheaves supported on sextic plane curves. Also in this case the author is able to give a complete classification. For any relevant value of \(\chi\), that is for \(0\leq \chi\leq 3\), he finds the natural stratification of the moduli space \(\mathrm{M}_{\mathbb{P}^2}(6,\chi)\). Every stratum is explicitely described in terms of locally free resolutions and cohomological conditions.