Maximal isotropic subbundles of orthogonal bundles of odd rank over a curve

Abstract

An orthogonal bundle over a curve has an isotropic Segre invariant determined by the maximal degree of a Lagrangian subbundle. This invariant and the induced stratifications on moduli spaces of orthogonal bundles, were studied for bundles of even rank in [4]. In this paper, we obtain analogous results for bundles of odd rank. We obtain a sharp upper bound on the isotropic Segre invariant. We show the irreducibility of the induced strata on the moduli spaces of orthogonal bundles of odd rank, and compute their dimensions. As a key ingredient of the proofs, we study the correspondence between Lagrangian subbundles of orthogonal bundles of even and odd rank.

Low rank orthogonal bundles and quadric fibrations

“…Furthermore, it is shown in [CH14] and [CH15] that w 2 (V ) coincides with the parity of the degree of any rank n isotropic subbundle of V , where n = rk (V )…”

“…This gives an analog for low rank orthogonal bundles of the enumeration results in [Hol04] for maximal degree subbundles of a general vector bundle, and in [CCH21a] for maximal degree Lagrangian subbundles of a general symplectic bundle. Some results in the paper are proven in or would follow easily from [CCH21b], [CH14] and [CH15]. However, our policy is, whenever possible, to give arguments which are elementary and/or use the rich and familiar geometry of the low rank situation.…”

“…Moreover, they are associated bundles of principal G-bundles for G an orthogonal group SO(r, C), O(r, C) or GO(r, C), and provide useful tools and intuition for questions regarding these principal bundles and their moduli. See for example [Bea08], [BG10], [CH14], [CH15], [Serm12], [CCH21b] and many other works.…”

“…In [CCH21b, § 2], it was shown that this is equivalent to the condition det V ∼ = L n (in general, the isomorphism V ∼ − → V * ⊗L only gives (det V ) 2 ∼ = L 2n ). Moreover, in [CH14] and [CH15] it was shown that when L = O C = det V , the invariant w 2 (V ) coincides with the parity of the degrees of the rank n isotropic subbundles of V . In Lemma 3.2 and Theorem 2.2 we offer more elementary proofs of these results.…”

Non-defectivity of Grassmannian bundles over a curve

“…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.…”