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Let's assume $X$ is a smooth algebraic surface and $C$ a curve containing a smooth point $p_0\in X$, then there exist divisors $H_1$ and $H_2$ non of which contain $p_0$ such that $C+H_1$ is linearly equivalent to $H_2$.

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This is a fairly straightforward consequence of the definition of ample divisor. It's probably useful to work out the details for yourself. – user5117 Feb 1 '12 at 17:23
Besides, it is worded like a homework problem, not as a question. – Angelo Feb 2 '12 at 4:40

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