= Abel map of a pointed smooth projective curve
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{title2=$\alpha(p)=\mathcal O(p-p_0)$}
Given a base point $p_0$, the Abel map takes points of a <smooth projective curve> to its <degree-zero Picard group of a curve>. For positive genus it is injective: equality at distinct $p,q$ would give a <principal divisor with one simple zero and one simple pole>, forcing a degree-one map to the <projective line> and genus zero. For genus one the <Riemann-Roch theorem> gives a unique effective degree-one representative of $L(p_0)$ for every degree-zero <line bundle> $L$, making the map bijective and inducing an <abelian group> law on the points.
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