## On a theorem of Castelnuovo. (Sur un théorème de Castelnuovo.)(French)Zbl 1139.14014

A birational map of the projective plane is called a Cremona transformation. If it preserves a pencil of lines, it is callled de Jonquières. The authors prove that $$F$$ is a Cremona transformation which fixes point-wise an irreducible curve of genus $$>1$$, then either $$F$$ is conjugate (in the Cremona group) to a de Jonquières transformation or it is of order 2 or 3. A slightly weaker version was proved by G. Castelnuovo [Rom. Acc. L. Rend. (5) I, No. 1, 47–50 (1892; JFM 24.0577.03); or see J. L. Coolidge, A treatise on algebraic plane curves, Dover (1959; Zbl 0085.36403)].

### MSC:

 14E07 Birational automorphisms, Cremona group and generalizations 14J26 Rational and ruled surfaces 14H50 Plane and space curves

### Citations:

Zbl 0085.36403; JFM 24.0577.03
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