Iitaka dimensions of vector bundles (Q1756425)

Let $E$ be a vector bundle on an irreducible projective variety $X$ (defined over an algebraically closed field). This vector bundle is asympotically generically generated if for some integer $m>0$ the $m$-th symmetric power $S^m E$ is globally generated on some nonempty open subset of $X$. Under this hypothesis one can define a rational map $\varphi_m$ from $X$ to the corresponding Grassmannian just using the evaluation map of global sections. Inspired on the case in which $E$ is a line bundle, the first result (see Theorem 3) on the paper under review is showing that for $m$ sufficiently large the rational maps $\varphi_m$ are birationaly equivalent to a fixed surjective morphism of projective varieties. This result slightly improves results in [\textit{E. Mistretta} and \textit{S. Urbinati}, ``Iitaka fibrations for vector bundles'', Int. Math. Res. Not. IMRN 2019, No. 7, 2223--2240 (2019; \url{doi:10.1093/imrn/rnx239})]. The dimension of the target of this fixed morphism is called the \textit{Iitaka dimension of} $E$ and denoted $k(X,E)$. The second result of this paper (see Thm. 4) provides a bound for this dimension in terms of the Iitaka dimension of the tautological bundle $\mathcal{O}(1)$ on the corresponding projective bundle $\mathbb{P}(E)$ and of the rank $r$ of $E$, to be precise: \[ k(X,E) \geq k(\mathbb{P}(E), \mathcal{O}(1))-r+1. \] In particular, for $E$ big, the Iitaka dimension of $E$ equals the dimension of $X$.