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If $X$ is Fano, that is, if $-K_X$ is ample, then (the closure of) the ample cone is polyhedral. This follows from Kleiman's criterion saying that it is dual to the cone of effective curves, and from Mori's Cone Theorem saying, roughly, that for any $\epsilon > 0$, the part of the effective cone where $-K_X > \epsilon$ is polyhedral. A convenient reference is http://en.wikipedia.org/wiki/Cone_of_curves. This is the case for the varieties $G/P$ discussed in other answers. The ample cone still need not be simplicial: toric varieties would furnish convenient counterexamples.

On the other hand, in general the ample cone need not be polyhedral at all. For example, if $C$ is an elliptic curve without complex multiplication, then the Néron-Severi group of $C \times C$ is generated by $C \times p$, $p \times C$, and the diagonal, and in terms of this basis the ample cone is given by $$a+b>0, \qquad b+c > 0, \qquad c+a>0, \quad\mbox{and} \quad ab + bc + ca >0: $$ a circular cone! See, for example, T. Bauer and C. Schulz, Seshadri constants on the self-product of an elliptic curve, J. Alg. 320 (2008) 2981-3005, Section 2.

If $X$ is Fano, that is, if $-K_X$ is ample, then (the closure of) the ample cone is polyhedral. This follows from Kleiman's criterion saying that it is dual to the cone of effective curves, and from Mori's Cone Theorem saying, roughly, that for any $\epsilon > 0$, the part of the effective cone where $-K_X > \epsilon$ is polyhedral. A convenient reference is http://en.wikipedia.org/wiki/Cone_of_curves. The varieties $G/P$ discussed in other answers are Fano. Even in the Fano case, the ample cone still need not be simplicial: toric varieties would furnish convenient counterexamples.

On the other hand, in general the ample cone need not be polyhedral at all. For example, if $C$ is an elliptic curve without complex multiplication, then the Néron-Severi group of $C \times C$ is generated by $C \times p$, $p \times C$, and the diagonal, and in terms of this basis the ample cone is given by $$a+b>0, \qquad b+c > 0, \qquad c+a>0, \quad\mbox{and} \quad ab + bc + ca >0: $$ a circular cone! See, for example, T. Bauer and C. Schulz, Seshadri constants on the self-product of an elliptic curve, J. Alg. 320 (2008) 2981-3005, Section 2.

If $X$ is Fano, that is, if $K_X^{-1}$ is ample, then (the closure of) the ample cone is polyhedral. This follows from Kleiman's criterion saying that it is dual to the cone of effective curves, and from Mori's Cone Theorem saying, roughly, that for any $\epsilon > 0$, the part of the effective cone where $K_X^{-1} > \epsilon$ is polyhedral. A convenient reference is http://en.wikipedia.org/wiki/Cone_of_curves. This is the case for the varieties $G/P$ discussed above. The ample cone still need not be simplicial: toric varieties would furnish convenient counterexamples.

On the other hand, in general the ample cone need not be polyhedral at all. For example, if $C$ is an elliptic curve without complex multiplication, then the Néron-Severi group of $C \times C$ is generated by $C \times p$, $p \times C$, and the diagonal, and in terms of this basis the ample cone is given by $$a+b>0, \qquad b+c > 0, \qquad c+a>0, \quad\mbox{and} \quad ab + bc + ca >0: $$ a circular cone! See, for example, T. Bauer and C. Schulz, Seshadri constants on the self-product of an elliptic curve, J. Alg. 320 (2008) 2981-3005, Section 2.

If $X$ is Fano, that is, if $-K_X$ is ample, then (the closure of) the ample cone is polyhedral. This follows from Kleiman's criterion saying that it is dual to the cone of effective curves, and from Mori's Cone Theorem saying, roughly, that for any $\epsilon > 0$, the part of the effective cone where $-K_X > \epsilon$ is polyhedral. A convenient reference is http://en.wikipedia.org/wiki/Cone_of_curves. This is the case for the varieties $G/P$ discussed in other answers. The ample cone still need not be simplicial: toric varieties would furnish convenient counterexamples.

On the other hand, in general the ample cone need not be polyhedral at all. For example, if $C$ is an elliptic curve without complex multiplication, then the Néron-Severi group of $C \times C$ is generated by $C \times p$, $p \times C$, and the diagonal, and in terms of this basis the ample cone is given by $$a+b>0, \qquad b+c > 0, \qquad c+a>0, \quad\mbox{and} \quad ab + bc + ca >0: $$ a circular cone! See, for example, T. Bauer and C. Schulz, Seshadri constants on the self-product of an elliptic curve, J. Alg. 320 (2008) 2981-3005, Section 2.

If $X$ is Fano, that is, if $K_X^{-1}$ is ample, then (the closure of) the ample cone is polyhedral. This follows from Kleiman's criterion saying that it is dual to the cone of effective curves, and from Mori's Cone Theorem saying, roughly, that for any $\epsilon$, the part of the effective cone where $K_X^{-1} > \epsilon$ is polyhedral. A convenient reference is http://en.wikipedia.org/wiki/Cone_of_curves. This is the case for the varieties $G/P$ discussed above. The ample cone still need not be simplicial: toric varieties would furnish convenient counterexamples.

On the other hand, in general the ample cone need not be polyhedral at all. For example, if $C$ is an elliptic curve without complex multiplication, then the Néron-Severi group of $C \times C$ is generated by $C \times p$, $p \times C$, and the diagonal, and in terms of this basis the ample cone is given by $$a+b>0, \qquad b+c > 0, \qquad c+a>0, \quad\mbox{and} \quad ab + bc + ca >0: $$ a circular cone! See, for example, T. Bauer and C. Schulz, Seshadri constants on the self-product of an elliptic curve, J. Alg. 320 (2008) 2981-3005, Section 2.

If $X$ is Fano, that is, if $K_X^{-1}$ is ample, then (the closure of) the ample cone is polyhedral. This follows from Kleiman's criterion saying that it is dual to the cone of effective curves, and from Mori's Cone Theorem saying, roughly, that for any $\epsilon > 0$, the part of the effective cone where $K_X^{-1} > \epsilon$ is polyhedral. A convenient reference is http://en.wikipedia.org/wiki/Cone_of_curves. This is the case for the varieties $G/P$ discussed above. The ample cone still need not be simplicial: toric varieties would furnish convenient counterexamples.

On the other hand, in general the ample cone need not be polyhedral at all. For example, if $C$ is an elliptic curve without complex multiplication, then the Néron-Severi group of $C \times C$ is generated by $C \times p$, $p \times C$, and the diagonal, and in terms of this basis the ample cone is given by $$a+b>0, \qquad b+c > 0, \qquad c+a>0, \quad\mbox{and} \quad ab + bc + ca >0: $$ a circular cone! See, for example, T. Bauer and C. Schulz, Seshadri constants on the self-product of an elliptic curve, J. Alg. 320 (2008) 2981-3005, Section 2.