The teacher is currently under surveillance in the hospital in the city of Grenoble.
A 16-year-old girl died on the spot in the avalanche at the Deux-Alpes ski resort, while a 14-year-old boy died later in hospital.
A Ukrainian man who was not part of the school group also died.
Grenoble prosecutor Jean-Yves Coquillat told a news conference that the investigation will seek to establish “the psychiatric state of the teacher” and whether he should have been leading a group of students.
He added that the ski run where the accident happened had been closed off with netting one metre high and 50 metres long.
There were also warnings in four different languages dotted around the run.
Despite the warnings, the group climbed over the netting “fully aware of what they were doing”, according to the prosecutor.
Reports in French daily Le Monde say 10 students have already been questioned by the police about the accident.
Another 30 will be interviewed in the coming days.
Le Monde also reports that several students on the morning before the accident expressed interest in skiing on the ski run, which has been closed since the beginning of the season, and that the teacher responsible for the ski trip had forbidden them to go onto the run.
But on the afternoon of accident 10 of the most experienced skiers among the students climbed over the netting and onto the run.
“They knew what they were doing,” Coquillat said.
It also emerged that several hundred other people had also used the ski run before the school students had climbed the fence.
One of the key objectives of the investigation, Coquillat added, will be to determine who started the avalanche.
Four other people - two Lithuanians, a Spaniard and a Czech - have died since the New Year in avalanches in the French Alps.