An [Enriques surface][1] (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.  

>> What are the ample and effective cones of an Enriques surface?

In particular, 

>> 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? 



  [1]: http://en.wikipedia.org/wiki/Enriques_surface