Seshadri constants on some Quot schemes

Let E be a vector bundle of rank n on ℙ 1 {\mathbb{P}^{1}} . Fix a positive integer d. Let 𝒬 ⁢ ( E , d ) {\mathcal{Q}(E,d)} denote the Quot scheme of torsion quotients of E of degree d and let Gr ⁢ ( E , d ) {\mathrm{Gr}(E,d)} denote the Grassmann bundle that parametrizes the d-dimensional quotients of the fibers of E. We compute Seshadri constants of ample line bundles on 𝒬 ⁢ ( E , d ) {\mathcal{Q}(E,d)} and Gr ⁢ ( E , d ) {\mathrm{Gr}(E,d)} .

Non-defectivity of Grassmannian bundles over a curve

Let [Formula: see text] be the Grassmann bundle of two-planes associated to a general bundle [Formula: see text] over a curve [Formula: see text]. We prove that an embedding of [Formula: see text] by a certain twist of the relative Plücker map is not secant defective. This yields a new and more geometric proof of the Hirschowitz-type bound on the isotropic Segre invariant for maximal isotropic sub-bundles of orthogonal bundles over [Formula: see text], analogous to those given for vector bundles and symplectic bundles in [I. Choe and G. H. Hitching, Secant varieties and Hirschowitz bound on vector bundles over a curve, Manuscripta Math. 133 (2010) 465–477, I. Choe and G. H. Hitching, Lagrangian sub-bundles of symplectic vector bundles over a curve, Math. Proc. Cambridge Phil. Soc. 153 (2012) 193–214]. From the non-defectivity, we also deduce an interesting feature of a general orthogonal bundle of even rank over [Formula: see text], contrasting with the classical and symplectic cases: a general maximal isotropic sub-bundle of maximal degree intersects at least one other such sub-bundle in positive rank.

Integrable spreads

SynopsisFor any integer k such that 0≦k≦m, Mk denotes the Grassmann bundle of tangent k-planes on the m-manifold M. A k-spread on M is a field Φ of tangent k-planes on Mk such that the derivative of the projection maps Φ(λ) to λ. Previous work by Douglas and others studied the local properties of such spreads. Here we develop the global theory, with special emphasis on the case in which Φ is integrable.