Non-defectivity of Grassmannian bundles over a curve

TL;DR

This paper proves the non-defectivity of Grassmannian bundles over a curve using a geometric approach, leading to new insights on the structure of orthogonal bundles and bounds on their invariants.

Contribution

It provides a new geometric proof of non-defectivity for Grassmannian bundles and derives implications for the intersection properties of maximal Lagrangian subbundles.

Findings

Embedding of Gr(2, E) is not secant defective.

New proof of Hirschowitz-type bound on Lagrangian Segre invariant.

Maximal Lagrangian subbundles intersect in positive rank.

Abstract

Let 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 Gr(2, E) by a certain twist of the relative Pl\"ucker map is not secant defective. This yields a new and more geometric proof of the Hirschowitz-type bound on the Lagrangian Segre invariant for orthogonal bundles over X, analogous to those given for vector bundles and symplectic bundles in [2, 3]. From the non-defectivity we also deduce an interesting feature of a general orthogonal bundle over X, contrasting with the classical and symplectic cases: Any maximal Lagrangian subbundle intersects at least one other maximal Lagrangian subbundle in positive rank.