The triple (C, c, m) is recoverable up to automorphism of the algebraic closure from the abstract group J_m(k-bar) and the embedded subset (C minus Supp(m))(k-bar).
Incidence systems on Cartesian powers of algebraic curves
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A curve and its abstract generalized Jacobian
The triple (C, c, m) is recoverable up to automorphism of the algebraic closure from the abstract group J_m(k-bar) and the embedded subset (C minus Supp(m))(k-bar).