A degree bound for globally generated vector bundles

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 , we obtain a sharp boundon 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 of degree at most N + 1. Research partially supported by the Spanish MCYT project MTM2006-04785 and by the program “Profesores de la UCM en el extranjero. Convocatoria 2006”.