Manin's book "Cubic forms" contains the calculations of these groups when $X$ is a smooth projective cubic surface. In particular, $Pic^0(X)={0}$ \operatorname{Pic}^0(X)={0}$and$Pic(X)=NS(X)$\operatorname{Pic}(X)=\operatorname{NS}(X)$ is a free commutative group of rank 7.
Another class of examples is provided by products $X=E \times E'$ of two elliptic curves. In particular, if $E=E'$ has no complex multiplication then $NS(E\times E)$ is a free commutative group of rank 3 generated by the classes of $E \times {0}$, ${0}\times E$ and the diagonal while $Pic^0(E\times \operatorname{Pic}^0(E\times E)=E \times E$. See Mumford's Abelian varietiesVarieties.
Manin's book "Cubic forms" contains the calculations of these groups when $X$ is a smooth projective cubic surface. In particular, $Pic^0(X)={0}$ and $Pic(X)=NS(X)$ is a free commutative group of rank 7.
Another class of examples is provided by products $X=E \times E'$ of two elliptic curves. In particular, if $E=E'$ has no complex multiplication then $NS(E\times E)$ is a free commutative group of rank 3 generated by the classes of $E \times {0}$, ${0}\times E$ and the diagonal while $Pic^0(E\times E)=E \times E$)E$. See Mumford's Abelian varieties. 1 Manin's book "Cubic forms" contains the calculations of these groups when$X$is a smooth projective cubic surface. In particular,$Pic^0(X)={0}$and$Pic(X)=NS(X)$is a free commutative group of rank 7. Another class of examples is provided by products$X=E \times E'$of two elliptic curves. In particular, if$E=E'$has no complex multiplication then$NS(E\times E)$is a free commutative group of rank 3 generated by the classes of$E \times {0}$,${0}\times E$and the diagonal while$Pic^0(E\times E)=E \times E\$). See Mumford's Abelian varieties.