An [Enriques surface](http://en.wikipedia.org/wiki/Enriques_surface) (in characteristic zero) is an algebraic surface which is the quotient of a K3 surface by a fixed-point-free involution. Such a surface has a rank 10 lattice of divisors. 



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> > (1) What are the ample and effective cones of an Enriques surface?
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In particular, 



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> > (2) Is it the case that there are ample divisors with arbitrarily large numbers of global sections which do not split nontrivially as the sum of two effective divisors? 
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**Edit:** As per Damiano's comment,



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> > (3) Is there *any* surface for which the answer to (2) is yes?
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What is known about the ample and effective cones of an Enriques surface?
The answer to question 2 is negative thanks to Kamawata-Morrison conjecture about nef cones of Calabi-Yau varieties that is proven for surfaces (a recent reference is here <http://arxiv.org/abs/1008.3825>), and to the comment of Damiano below.


Indeed, it is known that on a minimal two-dimesnional surface of Kodaira dimesnion 0 
the fundamental domain of the action of automorphism of the surface on the nef cone 
is rational polyhedral (see the above reference). So it is sufficient to consider only ample classes that belong to one rational polyhedral fundamental domain. Notice that every nef integral class (apart from $K$) on each Enriques surface is effective (see the proof of Damiano). Now notice that since the fundamental domain 
is rational polyhedral, the semi-group of integer vectors in it is finitely generated, so only finitely many integer vectors can not be presented as a sum of two others.
In other words, only finite number of ample divisors in a given fundamental domain are not linearly equivalent to a sum of several divisors. Since by definition all domains are the same, the statement is proven.


