Segre classes of a vector bundle spanned by global sections (Q1326550)

Let \(E\) denote a vector bundle on a projective variety \(X\) over a field \(k\), and let \(A_ * (X)\) denote the Chow group of \(X\). The Chern polynomial \(c_ t (E) \in A_ * (X) [t]\) and the Segre polynomial \(s_ t (E) \in A_ * (X)[t]\) determine one another by the relation \(c_ t (E)s_ t(E) = 1\). We shall prove in this note that if \(E\) is spanned by global sections then it is spanned by \(n + k\) of them, where, \(n = \text{rank} (E)\), \(k = \max [i | s_ i \neq 0]\) and \(s_ t (E) = 1 + s_ 1 t + s_ 2 t^ 2 + \cdots\). The kernel of the resulting epimorphism \((n + k) {\mathcal O} \to E \to 0\) provides a construction of a vector bundle of rank \(k\). We illustrate this method by `finding' two families of rank \(n - 1\) bundles on \(\mathbb{P}^ n_ k\) with `leading terms' being the zero correlation bundle [cf. \textit{R. Hartshorne}, Bull. Am. Math. Soc. 80, 1017-1032 (1974; Zbl 0304.14005)] and the Tango bundle [cf. \textit{H. Tango}, J. Math. Kyoto Univ. 16, 137-141 (1976; Zbl 0339.14008)].