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.

  • $\begingroup$ People do study the cone of effective divisors. Check out the book "Positivity in Algebraic Geometry I" by Lazarfeld. $\endgroup$ Jun 16, 2011 at 16:39
  • $\begingroup$ You are right, It is mentioned in definition 2.2.25 but there is not much discussion on it's structure. $\endgroup$ Jun 16, 2011 at 17:04
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    $\begingroup$ Let $X$ be a projective variety. By applying concept of Mori equivalence, projecteuclid.org/euclid.mmj/1030132722 , we can decompose the cone of effective divisors on $X$ into convex polyhedral chambers which called Mori chambers. In fact the best way to define a Mori dream space is if it has a good Mori chamber decomposition. $\endgroup$
    – user21574
    Jul 28, 2017 at 2:07

1 Answer 1


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:

  • 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:

  • 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.


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