The geometric Picard group

$X/k$ a K3 surface; $k\subset\Bbb{C}$ a number field.
$k^{\mathrm{al}}$ an algebraic closure of $k$, $X^{\mathrm{al}}:=X\times_k k^{\mathrm{al}}$.

$$\operatorname{Pic}(X^{\mathrm{al}}) \simeq \Z^{\rho}, \qquad \rho:=\rho(X^{\mathrm{al}})$$

$$\operatorname{Pic}(X^{\mathrm{al}}) \simeq \Z\langle \text{algebraic curves in } X^{\mathrm{al}} \rangle / \langle \text{linear equivalences} \rangle \subset H_2(X_{\Bbb{C}}, \Z)$$

Curves on $X^{\mathrm{al}}$ modulo linear equivalence.
For K3 surfaces $\Rightarrow$ linear/algebraic/numerical equivalence agree.
$\rho$, and more precisely the Picard lattice, is a coarse invariant.