摘要
Let X be a projective variety. If L is a line bundle on X, for each positive integer m in N(L) = { m∈ N∣ H 0 (X, L ⊗ m ) ≠ 0 } , the global sections of L ⊗ m define a rational map ϕm:X⤏Ym⊆P(H0(X,L⊗m)),where Y m is the closure of ϕ m (X). It is well-known that for all sufficiently large m∈ N(L) , the rational maps ϕ m : X⤏ Y m are birationally equivalent to a fixed fibration (the Iitaka fibration), and κ(L) : = dim Y m is called the Iitaka dimension of L. In a recent paper titled “Iitaka fibrations for vector bundles”, Mistretta and Urbinati generalized this to a vector bundle E on X. Let N(E) be the set of positive integers m such that the evaluation map H 0 (X, S m E) → S m E x is surjective for all points x in some nonempty open subset of X. For each m∈ N(E) , the global sections of S m E define a rational map φm:X⤏Ym⊆G(H0(X,SmE),rankSmE),where G(H 0 (X, S m E) , rank S m E) is the Grassmannian of rank S m E-dimensional quotients of H 0 (X, S m E). Mistretta and Urbinati showed that for every m∈ N(E) , the rational maps φ km are birationally equivalent for sufficiently large k, and called κ(E) : = dim Y km the Iitaka dimension of E. Here we first slightly improve Mistretta and Urbinati’s result to show that the rational maps φ m are birationally equivalent for all sufficiently large m∈ N(E). Then we show that κ(E)≥κ(OP(E)(1))-rankE+1.An immediate corollary of this inequality is that if E is big then κ(E) = dim X, which answers a question of Mistretta and Urbinati. Another corollary is that if E is big then det E is big, provided that N(E) ≠ ∅.