Nef line bundles on flag bundles on a curve over F?_p

摘要

Let E be a vector bundle of rank r over an irreducible smooth projective curve X defined over the field F?_p. For fixed integers r_1,...,r_ν with 1 ≤ r_1 <... < r_ν < r, let Fl(E) be the corresponding flag bundle over X associated to E. Let ξ → Fl(E) be a line bundle such that for every pair of the form (C,φ), where C is an irreducible smooth projective curve defined over F?_p and φ: C → Fl(E) is a nonconstant morphism, the inequality degree(φ*ξ) > 0 holds. We prove that the line bundle ξ is ample.