Maximal isotropic subbundles of orthogonal bundles of odd rank over a curve (Q2788774)

The Segre invariant of a vector bundle on a smooth projective curve is semicontinuous and hence gives a stratification of the corresponding moduli spaces. It was shown by \textit{B. Russo} and \textit{N. Teixidor i Bigas} [J. Algebr. Geom. 8, No. 3, 483--496 (1999; Zbl 0942.14013)] that the strata are irreducible and their dimension was computed. The analogous result for the isotropic Segre invariant of symplectic and orthogonal bundles of even rank was given by the authors [ibid. 25, No. 5, Article ID 1450047, 27 p. (2014; Zbl 1314.14061)].NEWLINENEWLINE The present paper deals with the same question for orthogonal bundles of odd rank. Let \(MO_X(2n+1)\) denote the moduli space of semistable orthogonal bundles of rank \(2n+1\) on a smooth projective curve of genus \(g\geq 2\). It consists of 2 connected components \(MO_X(2n+1)^\pm\).NEWLINENEWLINE The main result of the paper is the following theorem: For each even integer \(t\) with \(0< t<(n+ 1) (g-1)\), the stratum \(MO_X(2n+ 1,t)\) with isotropic Segre invaraint \(t\) is irreducible of dimension \({1\over 2}n(3n+ 1)(g- 1)+ {1\over 2}nt\). For \((n+1) (g-1)\leq t\leq(n+1) (g- 1)+3\) the stratum \(MO_X(2n+ 1,t)\) is dense in one of the components. This immediately gives a sharp upper bound for the isotropic Segre invariant.