The body of the victim, Joudia Zimmat, a 33-year-old mother of three of Tunisian origin, was found on scrubland where she regularly went jogging on Thursday.
Plant, who has been identified by family friends, lived with his mother 300 metres from the scene of the crime in a village near the historic city of Nimes. They moved there when his father, Denis, a telecoms executive who was half French, retired. Denis Plant died last year.
Her children’s school had alerted her partner after she failed to pick them up.
A box-cutter and two blood-stained shoes were found near the corpse and police described the killing as “slaughter”.
There was evidence that the victim had been sexually assaulted but not raped.
Police took Plant into custody after he proved “evasive” about his movements about the time of the murder.
They say that they were “surprised” by how much he corresponds to descriptions of someone seen near the scene and that he has deep scratches that could have been caused by brambles on his hands.
A search of his home led to the discovery of "quite a few objects that could be linked to the scene of the crime", local police chief Gilles Soulier told a press conference.
"He is not denying everything but he is not admitting everything either," Soulier told reporters.
Zimmat is the third female jogger to have been killed in France since June 2011.