Suppose $X$ is a surface, are there examples for curves on X which not 1-connected, but $h^0(C,O_C)=1$? (Here a curve is an effective divisor, not necessarily reduced or irreducible) (1-connectedness means when it splits into two effective divisors, the two parts have intesection number greater than or equal to 1)

Another question is I am not sure if the dimension of cohomology and 1-connectedness is preserved by linear equivalence ?