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Apr 5, 2017 at 13:32 history edited user41650 CC BY-SA 3.0
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Apr 5, 2017 at 13:29 comment added user41650 @potentiall dense, edited . If $X$ is say weak del pezzo surface of degree $K_X^2\geq 2$, then $|-K_X|$ is base point free, then the generic member of this linear system is smooth elliptic curve, in this case $H^0(O_C(D))\neq 0$.
Apr 5, 2017 at 12:12 comment added Lazzaro Campeotti I don't understand how "then it works" can be an answer to "what can we say about $X$?" Can you clarify exactly what the question is?
Apr 5, 2017 at 11:45 history edited user41650 CC BY-SA 3.0
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Apr 5, 2017 at 9:53 history edited user41650 CC BY-SA 3.0
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Apr 5, 2017 at 9:44 history asked user41650 CC BY-SA 3.0