On the Abel-Jacobi map for non-compact varieties (Q1382128)

Let \(X/\mathbb{C}\) be a smooth projective curve, \(S\) a reduced divisor on \(X\), \(J(X\setminus S)\) the generalized jacobian of \(X\setminus S\) and \(\text{Div}^0(X\setminus S)\) the set of divisors of degree \(0\) which do not intersect with \(S\). By integration one determines the Abel-Jacobi homomorphism \(\alpha:\text{Div}^0(X\setminus S)\to J(X\setminus S)\). The authors prove that \(\text{Ker} \alpha\) is the subgroup of \(S\)-principal divisors: \[ \text{Prin}_S(X)=\{(f)\in\text{Div}(X\setminus S):f\in K(X),\;f=1\text{ on }S\}. \] {}.