A lower bound for sectional genera of ample and spanned vector bundles on algebraic surfaces

Let E be an ample and spanned vector bundle of rank r over a complex projective surface S. It is shown that the sectional genus g(S, det E) is bounded from below by the number b = max{q(S), 2 - rchi(O(S)))} and pairs (S, E) satisfying b less-than-or-equal-to g(S, det E) less-than-or-equal-to b + 1 are characterized.