Suppose $A,B$ are curves on smooth projective surface, having no common components and intersect, so $(A.B)>0$,do we have $H^0(O_A(-B|_A))=0?$ (here $A,B$ are effective divisors, may be irreduced and reducible)
If $A$ is irreducible it is right since any regular section associates to a divisor of positive degree, and the degree is preserved by linear equivalence. But in this case it seemes not to be the concept of linear equivalence preserves degree.
I think there may be counter examples although it seems right.