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Timeline for What does the ample cone look like?

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Dec 2, 2010 at 10:52 comment added user5117 ...because there is only one extremal ray in $K_X^{<0}$. But the cone of curves is not polyhedral, i.e. not generated by a finite set of rays.
Dec 2, 2010 at 10:49 comment added user5117 Dear Moon, the point (a little late) is that there are only finitely many extremal rays in any of the half-spaces where -K is less than some chosen $\epsilon$, but there may be infinitely (even uncountably) many altogether. To visualise this, imagine taking a transverse "slice" of the cone of curves (as in the picture on p.60 of Lazarsfeld Vol. 1). Suppose that $rho(X)=4$, so this slice lives in 3-space. Then we could have the following picture: this slice is the convex set generated by a circle in $K_X^\perp$ together with a single point in $K_X^{<0}$. This doesn't violate the Cone Theorem...
Nov 5, 2010 at 1:43 comment added Moon To Michael Thaddeus, Thank you for a nice answer and comment, but I can't imagine that cone with finitely many extremal rays may not be polyhedral. Is there a counterexample or am I missing some obvious picture?
Jul 13, 2010 at 15:52 comment added Michael Thaddeus Rob Lazarsfeld pointed out an inaccuracy in my remarks above. Mori's Cone Theorem does not say that the part of the effective cone where -K exceeds epsilon is polyhedral. Rather, it says that the effective cone is generated by the part where -K is less than epsilon, together with finitely many additional rays. This implies the statement of the previous sentence when the Neron-Severi group has rank ≤ 3, but not otherwise. It also implies that the ample cones of Fano varieties are polyhedral, as stated.
Jun 8, 2010 at 8:54 history edited Michael Thaddeus CC BY-SA 2.5
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Jun 7, 2010 at 21:55 comment added Michael Thaddeus oops, sorry, fixed.
Jun 7, 2010 at 21:53 history edited Michael Thaddeus CC BY-SA 2.5
changed to additive notation!
Jun 7, 2010 at 15:11 history edited Michael Thaddeus CC BY-SA 2.5
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Jun 7, 2010 at 11:49 history edited Michael Thaddeus CC BY-SA 2.5
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Jun 7, 2010 at 11:43 history edited Michael Thaddeus CC BY-SA 2.5
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Jun 7, 2010 at 11:33 comment added user5117 Tiny quibble: I guess "the part of the effective cone where K_X > epsilon is polyhedral" should be "the part of the effective cone where -K_X > epsilon is polyhedral".
Jun 7, 2010 at 10:50 history answered Michael Thaddeus CC BY-SA 2.5