Cone of effective divisors!

2011-06-16T15:37:33Z

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.

Answer by Dave Anderson for Cone of effective divisors!

2011-06-16T18:53:55Z

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.

Cone of effective divisors! - MathOverflow

Cone of effective divisors!

Answer by Dave Anderson for Cone of effective divisors!