COURBES DE GENRE 5 MUNIES D’UNE INVOLUTION SANS POINT FIXE

We study genus 5 curves C with a fixed point free involution. We give a geometrical (embedded) characterisation of these curves among all genus 5 curves : the points of the Prym variety associated to the involution give embeddings of the curve C in P3 so that C has infinitely many quadrisecant lines. Conversely, any genus 5 curve having such an embedding is endowed with a fixed point free involution and the embedding is given by a point of the Prym variety.