Narrow Results

Let $\mathrm{\text{Gr}}(2,E)$ be the Grassmann bundle of two-planes associated to a general bundle $E$ over a curve $X$. We prove that an embedding of $\mathrm{\text{Gr}}(2,E)$ by a certain twist of the relative Plücker map is not secant defective. This yields a new and more geometric proof of the Hirschowitz-type bound on the isotropic Segre invariant for maximal isotropic sub-bundles of orthogonal bundles over $X$, analogous to those given for vector bundles and symplectic bundles in [I. Choe and G. H. Hitching, Secant varieties and Hirschowitz bound on vector bundles over a curve, Manuscripta Math.133 (2010) 465–477, I. Choe and G. H. Hitching, Lagrangian sub-bundles of symplectic vector bundles over a curve, Math. Proc. Cambridge Phil. Soc.153 (2012) 193–214]. From the non-defectivity, we also deduce an interesting feature of a general orthogonal bundle of even rank over $X$, contrasting with the classical and symplectic cases: a general maximal isotropic sub-bundle of maximal degree intersects at least one other such sub-bundle in positive rank.

We study various aspects on nontrivial logarithmic co-Higgs structure associated to unstable bundles on algebraic curves. We check several criteria for (non-)existence of nontrivial logarithmic co-Higgs structures and describe their parameter spaces. We also investigate the Segre invariants of these structures and see their non-simplicity. In the end, we also study the higher dimensional case, specially when the tangent bundle is not semistable.

The aim of this paper is to generalize the $m$-Segre invariant for vector bundles to coherent systems. Let $X$ be a non-singular irreducible complex projective curve of genus $g\ge 0$ and $G(\alpha ;n,d,k)$ be the moduli space of $\alpha$-stable coherent systems of type $(n,d,k)$ on $X$. For any pair of integers $(m,t)$ with $0<m<n$, $0\le t\le k$ we define the $(m,t)$-Segre invariant, and prove that it defines a lower semicontinuous function on the families of coherent systems. Thus, the $(m,t)$-Segre invariant induces a stratification of the moduli space $G(\alpha ;n,d,k)$ into locally closed subvarieties $G(\alpha ;n,d,k;m,t;s)$ according to the value $s$ of the function. We determine an above bound for the $(m,t)$-Segre invariant and compute a bound for the dimension of the different strata $G(\alpha ;n,d,k;m,t;s)$. Moreover, we give some conditions under which the different strata are nonempty. To prove the above results, we introduce the notion of coherent systems of subtype $(a)$.