Linear series on surfaces and Zariski decomposition

Remark 1. We quickly recall a couple of definitions Let DivQ(X) := Div(X)⊗Q. On smooth projective surfaces all Q-Weil divisors are also Q-Cartier, hence we can write Q-Cartier every divisor D as a finite sum ∑ xiCi, where the Ci are distinct integral curves and xi ∈ Q. A divisor D is called effective (or sometimes positive) if xi ≥ 0 ∀i. If D · C ≥ 0 for all integral curves C then D we be called nef. For a divisor D = ∑ xiCi we define the intersection matrix to by the symmetric matrix I(D) = I(C1, . . . , Cq) = (Ci ·Cj)i,j. By definition we have D ·D = xt · I(D) · x, where x = (x1, . . . , xn). We will call P a sub divisor of D, if D − P is effective or zero. We will then write P D.