= Rational points need not generate the degree-zero Picard group
Let $C$ be the smooth projective model over $\mathbb F_2$ of $y^2+y=x^5+x+1$. Its only rational point is its unique point $p_\infty$ at infinity. At that point $x,y$ have pole orders two and five. The degree-two closed point $D$ defined by $y=0$, $x^2+x+1=0$ gives a nonzero class $[D-2p_\infty]$. Indeed a function with poles bounded by $2p_\infty$ has the form $a+bx$: in the affine coordinate ring every function is uniquely $A(x)+B(x)y$, and the two possible pole orders have opposite parity. No nonzero $a+bx$ vanishes at both roots of $x^2+x+1$. Thus the <Abel map of a pointed smooth projective curve> on rational points has trivial image here, but the <degree-zero Picard group of a curve> is nontrivial. Generation by point differences needs an algebraically closed field, or closed points with their residue degrees instead of just rational points.
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