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  1. ; ; (, Compositio Mathematica)
    Let$$X$$be a smooth geometrically connected projective curve over the field of fractions of a discrete valuation ring$$R$$, and$$\mathfrak {m}$$a modulus on$$X$$, given by a closed subscheme of$$X$$which is geometrically reduced. The generalized Jacobian$$J_\mathfrak {m}$$of$$X$$with respect to$$\mathfrak {m}$$is then an extension of the Jacobian of$$X$$by a torus. We describe its Néron model, together with the character and component groups of the special fibre, in terms of a regular model of$$X$$over$$R$$. This generalizes Raynaud's well-known description for the usual Jacobian. We also give some computations for generalized Jacobians of modular curves$$X_0(N)$$with moduli supported on the cusps. 
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