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I'd say Lazarsfeld's book "Positivity in algebraic geometry I,II" is the standard reference these days. In particular, Volume I has a lot of explicit examples. I also recommend Debarre's book 'Higher dimensional algebraic geometry" which is similar in style with a lot of nice examples and explicit computations.

If you are familiar with toric geometry, there is (not surprisingly) a simple description of the psedudoeffective cone in terms of the combinatorial data in the fan. See Cox-Little-Schenck's new book 'Toric varieties' for details. This gives a hoard of interesting examples.

Example. Let $X$ be the blow-up of $\mathbb{P}^2$ at two points and let $E_1,E_2$ be the exceptional divisors. A basis for $Pic(X)$ is given by $L,E_1,E_2$ where $L$ is the pull back of a general line in $\mathbb{P}^2$. We show that $\overline{Eff}(X)$ is spanned by $E_1,E_2$ and the strict transform of the line $L_0=L-E_1-E_2$. Let $\tau$ be the cone spanned by these three classes. Since they are all effective we have $\tau\subset \overline{Eff}(X)$. Coversely, let $D$ be any effective divisor with class $aL+bE_1+cE_2$. We will show that $D$ can be written as a sum of elements from $\tau$. We may assume $D$ to be irreducible. If $D$ is not one of the $E_1,E_2,L_0$, we then have $D.E_i\ge 0$ and $D.L_0\ge 0$. In particular, $D$ belongs to the dual cone of $\tau$, which is easily computed as $\tau^*=\langle L,L-E_1,L-E_2\rangle_{\ge 0}$. Now $L, L-E_1, L-E_2$ are all effective and can be written as positive linear combinations of $E_1,E_2,L_0$, and hence so can $D$. As a by-product, we have just computed the nef cone, which is $\tau^\*$.

B. Harbourne "Global aspects of the geometry of surfaces" and

Y. Tschikel "Algebraic varieties with many rational points.

For K3 surfaces, S. Kovacs has a nice paper on the 'Cone of curves of a K3 surface' (see also this answer). There are also many explicit examples in Artebani-Hausen-Laface's paper On Cox rings of K3-surfaces.

I can also recommend Artie Prendergast-Smith's papers at his homepage. In particular, his PhD thesis contains a very explicit example where he computes the of the effective cone of a rational threefold.

A. Laface, M. Velasco, A survey on Cox rings

I. Arzhantsev, U. Derenthal, J. Hausen, A. Laface, Cox rings

J. Gonzalez, M. Hering, S. Payne, H. Süß Cox rings and pseudoeffective cones of projectivized toric vector bundles and

M. Artebani, A. Laface Cox rings of surfaces and the anticanonical Iitaka dimension

I'd say Lazarsfeld's book "Positivity in algebraic geometry I,II" is the standard reference these days. In particular, Volume I has a lot of explicit examples. I also recommend Debarre's book 'Higher dimensional algebraic geometry" which is similar in style with a lot of nice examples and explicit computations.

If you are familiar with toric geometry, there is (not surprisingly) a simple description of the psedudoeffective cone in terms of the combinatorial data in the fan. See Cox-Little-Schenck's new book 'Toric varieties' for details. This gives a hoard of interesting examples.

Example. Let $X$ be the blow-up of $\mathbb{P}^2$ at two points and let $E_1,E_2$ be the exceptional divisors. A basis for $Pic(X)$ is given by $L,E_1,E_2$ where $L$ is the pull back of a general line in $\mathbb{P}^2$. We show that $\overline{Eff}(X)$ is spanned by $E_1,E_2$ and the strict transform of the line $L_0=L-E_1-E_2$. Let $\tau$ be the cone spanned by these three classes. Since they are all effective we have $\tau\subset \overline{Eff}(X)$. Conversely, let $D$ be any effective divisor with class $aL+bE_1+cE_2$. We will show that $D$ can be written as a sum of elements from $\tau$. We may assume $D$ to be irreducible. If $D$ is not one of the $E_1,E_2,L_0$, we then have $D.E_i\ge 0$ and $D.L_0\ge 0$. In particular, $D$ belongs to the dual cone of $\tau$, which is easily computed as $\tau^*=\langle L,L-E_1,L-E_2\rangle_{\ge 0}$. Now $L, L-E_1, L-E_2$ are all effective and can be written as positive linear combinations of $E_1,E_2,L_0$, and hence so can $D$. As a by-product, we have just computed the nef cone, which is $\tau^*$.

B. Harbourne "Global aspects of the geometry of surfaces" and

Y. Tschikel "Algebraic varieties with many rational points.

For K3 surfaces, S. Kovacs has a nice paper on the 'Cone of curves of a K3 surface' (see also this answer). There are also many explicit examples in Artebani-Hausen-Laface's paper On Cox rings of K3-surfaces.

I can also recommend Artie Prendergast-Smith's papers at his homepage. In particular, his PhD thesis contains a very explicit example where he computes the of the effective cone of a rational threefold.

A. Laface, M. Velasco, A survey on Cox rings

I. Arzhantsev, U. Derenthal, J. Hausen, A. Laface, Cox rings

J. Gonzalez, M. Hering, S. Payne, H. Süß Cox rings and pseudoeffective cones of projectivized toric vector bundles and

M. Artebani, A. Laface Cox rings of surfaces and the anticanonical Iitaka dimension

For K3 surfaces, S. Kovacs has a nice paper on the 'Cone of curves of a K3 surface' (see also this answer). There are also many explicit examples in Artebani-Hausen-Laface's paper On Cox rings of K3-surfaces.

For K3 surfaces, S. Kovacs has a nice paper on the 'Cone of curves of a K3 surface' (see also this answer). There are also many explicit examples in Artebani-Hausen-Laface's paper On Cox rings of K3-surfaces.

If you are familiar with toric geometry, there is (not surprisingly) a simple description of the psedudoeffective cone in terms of the combinatorial data in the fan. See Cox-Little- Schenk's book 'Toric varieties' chapter 4 for details. This gives a lot of examples.

Example. Let $X$ be the blow-up of $\mathbb{P}^2$ at two points and let $E_1,E_2$ be the exceptional divisors. A basis for $Pic(X)$ is given by $L,E_1,E_2$ where $L$ is the pull back of a general line in $\mathbb{P}^2$. We show that $\overline{Eff}(X)$ is spanned by $E_1,E_2$ and the strict transform of the line $L_0=L-E_1-E_2$. Let $\tau$ be the cone spanned by these three classes. Since they are all effective we have $\tau\subset \overline{Eff}(X)$. Coversely, let $D$ be any effective divisor with class $aL+bE_1+cE_2$. We will show that $D$ can be written as a sum of elements from $\tau$. We may assume $D$ to be irreducible. If $D$ is not one of the $E_1,E_2,L_0$, we then have $D.E_i\ge 0$ and $D.L_0\ge 0$. In particular, $D$ belongs to the dual cone of $\tau$, which is easily computed as $\langle L,L-E_1,L-E_2\rangle_{\ge 0}$. Now $L, L-E_1, L-E_2$ are all effective (since they are clearly the strict transforms of effective curves in $\mathbb{P}^2$), hence $D$ can be written as a positive sum of elements of $\tau$. As a by-product, we have just computed the nef cone, which is $\tau^\*$.

A. Laface, M. Velasco, A survey on Cox rings

J. Hausen, Three lectures on Cox rings

If you are familiar with toric geometry, there is (not surprisingly) a simple description of the psedudoeffective cone in terms of the combinatorial data in the fan. See Cox-Little-Schenck's new book 'Toric varieties' for details. This gives a hoard of interesting examples.

Example. Let $X$ be the blow-up of $\mathbb{P}^2$ at two points and let $E_1,E_2$ be the exceptional divisors. A basis for $Pic(X)$ is given by $L,E_1,E_2$ where $L$ is the pull back of a general line in $\mathbb{P}^2$. We show that $\overline{Eff}(X)$ is spanned by $E_1,E_2$ and the strict transform of the line $L_0=L-E_1-E_2$. Let $\tau$ be the cone spanned by these three classes. Since they are all effective we have $\tau\subset \overline{Eff}(X)$. Coversely, let $D$ be any effective divisor with class $aL+bE_1+cE_2$. We will show that $D$ can be written as a sum of elements from $\tau$. We may assume $D$ to be irreducible. If $D$ is not one of the $E_1,E_2,L_0$, we then have $D.E_i\ge 0$ and $D.L_0\ge 0$. In particular, $D$ belongs to the dual cone of $\tau$, which is easily computed as $\tau^*=\langle L,L-E_1,L-E_2\rangle_{\ge 0}$. Now $L, L-E_1, L-E_2$ are all effective and can be written as positive linear combinations of $E_1,E_2,L_0$, and hence so can $D$. As a by-product, we have just computed the nef cone, which is $\tau^\*$.

A. Laface, M. Velasco, A survey on Cox rings