We continue the study of G. Castelnuovo on the group of birational

transformation of the complex plane that fix each point of a curve of genus

> 1 ; we use adjoint linear system of the curve as Castelnuovo does.

We prove that these groups are abelian, and that these are either finite, of

order 2 or 3, or conjuguate to a subgroup of the de Jonqui`eres group. We show

also that these results do not generalise to curves of genus  1.

Keywords. Cremona transformations, birational transformations, fixed curves,

curves of big genus, adjoint linear system, de Jonqui`eres transformations.