A stratification on the moduli spaces of symplectic and orthogonal bundles over a curve

Abstract

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.

Non-defectivity of Grassmannian bundles over a curve

“…Lange and Newstead showed in [10,Proposition 2.4] that if W is a generic vector bundle of rank r and if k ≤ r/2, then two maximal subbundles of rank k in W intersect generically in rank zero. The analogous statement for maximal Lagrangian subbundles of a generic symplectic bundle was proven in [4,Theorem 4.1 (3)]. However, as noted in [4,Remark 5.1], the corresponding approach in the orthogonal case does not exclude the possibility that two maximal Lagrangian subbundles intersect in a line bundle.…”

“…In [4, Theorem 1.3 (1)], a sharp upper bound on the value of t(V ) was given, based on the computation of the dimensions of certain Quot schemes. In this section we use Theorem 3.1 together with a lifting criterion from [4] to give a more geometric proof of this upper bound.…”

“…In Theorem 4.3, the density of Sec k Gr(2, F ) in PH 1 (X, ∧ 2 F ) is used to produce a lower bound on the degrees of maximal Lagrangian subbundles of a general orthogonal extension 0 → E → V → E * → 0. On the other hand, in [4,Theorem 5.2], the fact that certain Sec k Gr(2, E) are not dense in their respective PH 1 (X, ∧ 2 E) is used to give upper bounds on the degrees of maximal Lagrangian subbundles of the corresponding extensions.…”

“…Furthermore, in §5 we use Theorem 3.1 to answer a question which we were unable to solve with the methods in [4]: We show that any maximal Lagrangian subbundle of a general orthogonal bundle meets another maximal Lagrangian subbundle in a sheaf of positive rank. In this way orthogonal bundles behave differently from general vector bundles and symplectic bundles.…”

“…Although the present note can be read independently of [2,3,4,5], we use several results from these articles. In particular, access to [4, §2 and §5] may be helpful for the reader.…”