Let $X$ be a weak del pezzo surface, I Wonder whether the following statment is true:
Let $L$ be a line bundle on $X$, then $h^0(L)=0$ implies $h^0(nL)=0$ for all $n\geq 1$.
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$\begingroup$ If $X$ is assumed to be del Pezzo, this is okay, right? $\endgroup$– Chen JiangCommented Mar 15, 2017 at 13:50
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$\begingroup$ @gbp Thanks, could you turn your comments on del pezzo surface case into an answer? I dont see how the "generators are primitive" will imply this statement $\endgroup$– user41650Commented Mar 15, 2017 at 21:34
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$\begingroup$ @gbp We know that the effective cone is generated by (-1) and (-2)-curves (d≤7) from arxiv.org/pdf/math/0703202.pdf (Theorem 3.10) The paper arxiv.org/pdf/math/0604194.pdf gives generators of Cox rings and hence generators of effective monoids. However, I don't see what the question has to do with primitivity of generators. The group generated by two primitive vectors in Z^2 need not be saturated, and even when the group is saturated, the monoid they generate need not be saturated (e.g. (0,1) doesn't lie in the monoid generated by (1,0) and (-1,2)). $\endgroup$– Junyan XuCommented Mar 15, 2017 at 21:35
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$\begingroup$ I still think that it can be effectively determined whether a finitely generated monoid is saturated given its generators, but I don't know about an algorithm. Maybe people working in toric varieties know where such an algorithm is implemented... $\endgroup$– Junyan XuCommented Mar 15, 2017 at 21:35
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$\begingroup$ A side question is whether the effective monoid is also generated by (-1) and (-2)-curves; this is false for degree 1 genuine del Pezzo surfaces, but I am yet to see another counterexample. $\endgroup$– Junyan XuCommented Mar 15, 2017 at 21:49
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1 Answer
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Jun Yan found a counter example for weak del pezzo surface.
Let $X$ be weak del pezzo surface of degree $4$, Let $X=X_{4,4A_1}$, irreducible $(-2)$-curves are $E_1-E_2,L_{123}=L-E_1-E_2-E_3,E_4-E_5,L_{345}=L-E_3-E_4-E_5$, take sum of all these curves, we get:$2(L_{235})$ which is an effective divisor. But $L_{235}$ itself is not effective divisor. Which means that rationally effective divisor is not effective. And actually, there are a series such examples
But as Chen Jiang pointed out in the comment, the statement is true for del pezzo surface.