Cone of effective divisors! Let $X$ be a smooth simply connected projective variety of dimension $n$ (over complex numbers of course). For such $X$ we have two famous cones which are cone of effective curves and ample cone and are dual to each other.
Question: Is there any thing as Cone of effective divisors? Is there any problem to define such a thing? Has any body studied that?
For surfaces, it is just cone of effective curves. So the smallest dimension at which we would get some thing new is three.
 A: As mentioned in the comments, the (pseudo)effective cone $\overline{\mathrm{Eff}}(X)$, defined as the closure of the cone of all effective divisors on $X$, is certainly an object of study, and Lazarsfeld's book is a good reference.  Your complaint that he doesn't say much about its structure is surely related to the fact that so little is known!  Here are a few general things I'm aware of:


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*The interior of the effective cone is the big cone, i.e., the cone of line bundles with positive volume.

*The dual of the effective cone is the cone of moveable curves, see Boucksom-Demailly-Paun-Peternell.

*As part of their work on the minimal model program, Birkar-Cascini-Hacon-McKernan prove that log Fano varieties have finitely generated effective cones.
And here are a couple specific instances where one knows more:


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*When $X$ admits an action by a solvable group with a dense orbit, the effective cone is generated by the components of the complement of the orbit.  (This works when $X$ is, e.g., a toric variety or a Schubert variety.)

*There's been a lot of recent work on the case $X=\overline{M}_{0,n}$, see e.g., Hu-Keel, Hassett-Tschinkel, Castravet-Tevelev.
