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 C be a complex projective smooth curve and W a symplectic vector bundle of rank 2n over C. The Lagrangian Quot scheme LQ −e (W ) parameterizes subsheaves of rank n and degree −e which are isotropic with respect to the symplectic form. We prove that LQ −e (W ) is irreducible and generically smooth of the expected dimension for all large e, and that a generic element is saturated and stable. The proof relies on the geometry of symplectic extensions.

Let X be a non-singular complex projective curve of genus ≥ 3. Choose a point x ∈ X . Let M x be the moduli space of stable bundles of rank 2 with determinant O X (x). We prove that the Chow group CH Q 1 (M x ) of 1-cycles on M x with rational coefficients is isomorphic to CH Q 0 (X ). By studying the rational curves on M x , it is not difficult to see that there exits a natural homomorphism CH 0 (J ) → CH 1 (M x ) where J denotes the Jacobian of X . The crucial point is to show that this homomorphism induces a homomorphism CH 0 (X ) → CH 1 (M x ), namely, to go from the infinite dimensional object CH 0 (J ) to the finite dimensional object CH 0 (X ). This is proved by relating the degeneration of Hecke curves on M x to the second term I * 2 of Bloch's filtration on CH 0 (J ).