Singular Points of Plane Curves

ly isomorphic to (C×)r−1 × (C), and hence also to (S1)r−1 × (R), where r = |J | is the number of branches and k = δ(C)− r+1 = 1 2 (μ(C) + 1 − r). The construction of the Jacobian variety J(C̃) of the non-singular curve C̃ in the large is standard in algebraic geometry. There is also a notion of Jacobian of a singular curve C , defined e.g. in [85], which, like the other, is an abelian group. There is a natural exact sequence 0 → O× C → O× J → J(C) → J(C̃) → 0, thus the kernel of the (split) surjection J(C) → J(C̃) can be identified with the direct product over singular points P of C of the local Jacobians.