A degree bound for globally generated vector bundles

Abstract

Let E be a globally generated vector bundle of rank e ≥ 2 over a reduced irreducible projective variety X of dimension n defined over an algebraically closed field of characteristic zero. Let L := det(E). If deg(E) := deg(L) = L n > 0 and E is not isomorphic to $${\mathcal{O}_X^{e-1}\oplus L}$$ , we obtain a sharp bound $${\rm deg}(E)\geq h^0(X,E)-e$$ on the degree of E, proving also that deg(L) = h 0(X, L) − n if equality holds. As an application, we obtain a Del Pezzo–Bertini type theorem on varieties of minimal degree for subvarieties of Grassmannians, as well as a bound on the sectional genus for subvarieties $${X \subset \mathbb{G}(k,N)}$$ of degree at most N + 1.