Downsides of combining register / login with e-mail recognition

I'm probably not the first with this idea but I don't know the right terms to find an article/topic on this.

My idea is to have one central login/registration place starting with only an e-mail field, then when a user has filled in their e-mail, different options appear based on whether or not the e-mail is already registered.

I'm curious what would be the downside(s) of this approach.