Let (S, L) be a polarized surface and let π:˜{S}→ S be the blow-up at r points p
1, ..., p
r on S. Set \displaystyle ˜{L}=π
*L-∑ a
iE
i, where a
i's are positive integers and E
i's are the (-1)-curves over p
i. We consider whether ˜{L} is ample or not if p
i's are in a general position. The cases of sectional genus two are studied especially precisely.
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