Abstract. For a vector bundle V over a curve X of rank n and for each integer r in the range 1 ≤ r ≤ n − 1, the Segre invariant sr is defined by generalizing the minimal self-intersection number of the sections on a ruled surface. In this paper we generalize Lange and Narasimhan's results on rank 2 bundles which related the invariant s1 to the secant varieties of the curve inside certain extension spaces. For any n and r, we find a way to get information on the invariant sr from the secant varieties of certain subvariety of a scroll over X. Using this geometric picture, we obtain a new proof of the Hirschowitz bound on sr.

Abstract. A symplectic or orthogonal bundle V of rank 2n over a curve has an invariant t(V ) which measures the maximal degree of its isotropic subbundles of rank n. This invariant t defines stratifications on moduli spaces of symplectic and orthogonal bundles. We study this stratification by relating it to another one given by secant varieties in certain extension spaces.We give a sharp upper bound on t(V ), which generalizes the classical Nagata bound for ruled surfaces and the Hirschowitz bound for vector bundles, and study the structure of the stratifications on the moduli spaces. In particular, we compute the dimension of each stratum. We give a geometric interpretation of the number of maximal Lagrangian subbundles of a general symplectic bundle, when this is finite. We also observe some interesting features of orthogonal bundles which do not arise for symplectic bundles, essentially due to the richer topological structure of the moduli space in the orthogonal case.

AbstractA symplectic bundle over an algebraic curve has a natural invariantsLagdetermined by the maximal degree of its Lagrangian subbundles. This can be viewed as a generalization of the classical Segre invariants of a vector bundle. We give a sharp upper bound onsLagwhich is analogous to the Hirschowitz bound on the classical Segre invariants. Furthermore, we study the stratifications induced bysLagon moduli spaces of symplectic bundles, and get a full picture for the case of rank four.

Let V be a vector bundle over a smooth curve C. In this paper, we study twisted Brill–Noether loci parametrising stable bundles E of rank n and degree e with the property that $$h^0 (C, V \otimes E) \ge k$$ h 0 ( C , V ⊗ E ) ≥ k . We prove that, under conditions similar to those of Teixidor i Bigas and of Mercat, the Brill–Noether loci are nonempty and in many cases have a component which is generically smooth and of the expected dimension. Along the way, we prove the irreducibility of certain components of both twisted and “nontwisted” Brill–Noether loci. We describe the tangent cones to the twisted Brill–Noether loci. We end with an example of a general bundle over a general curve having positive-dimensional twisted Brill–Noether loci with negative expected dimension.

We review the notions of symplectic and orthogonal vector bundles over curves, and the connection between principal parts and extensions of vector bundles. We give a criterion for a certain extension of rank 2n to be symplectic or orthogonal. We then describe almost all of its rank n vector subbundles using graphs of sheaf homomorphisms, and give criteria for the isotropy of these subbundles. Finally, we sketch the use of these ideas in moduli questions for symplectic vector bundles.