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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.
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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.
show/hide this revision's text 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.