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 ?



Examples for curve not 1-connected but $h^0(C, O_C)=1$
Here is an example with $X$ smooth : consider a regular surface $X$ with an elliptic fibration having a double fiber $2E$ (typically, an Enriques surface). Then $2E=E+E$ is not 1-connected ($E^2=0$), but one checks easily that $H^1(X,\mathcal{O}\_X(-2E))=0$, hence $h^0(\mathcal{O}\_{2E})=1$.


