Suppose $X$ is a surface, are there examples for curves on X which not 1-connected, but $h^0(C)=1$? (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)
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Examples for curve not 1-connected but $h^0(C)=1$
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