Downsides of combining register / login with e-mail regocnistion

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 / register place starting with only an e-mail field, then when a user has filled in their e-mail, options appear based on if the e-mail is already registered or not.

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