  We give optimal effective bounds for some well-known theorems on complex
algebraic surfaces, which are respectively due to Serre, Zariski (1962),
Castelnuovo (1897), Artin (1962, 1966), Benveniste (1984), Cutkosky and
Srinivas (1993). These theorems are about Riemann-Roch problem (on the behavior
of the function dim |nD| of n), vanishing theorems, base-point freeness and
k-very ampleness of the linear systems |nD| and |nA+L|, where D is effective, A
is nef and big and L is arbitrary. As a consequence, we obtain an effective
version of Matsusaka's big theorem, and we give also examples to show that our
bound is the best possible one.
