First things first - what you're saying applies only to highly functional/traditional organizations.
Now let me answer your question:
1- Neither the project manager, nor the line manager, have authority over the stakeholders. Stakeholders are usually company executives and managers that (often far) outrank the project manager and the line manager. Usually stakeholders have authority over the project/line manager and not the other way around. (unless the line manager is a stakeholder himself, which can happen, in this case some stakeholders may not have authority over the line manager)
2- The project manager is technically not a management role, and that's why the project manager doesn't have authority over the resources. The project manager's job is to facilitate the project through all its stages - in the executing stage this facilitation is about allocating tasks to resources and monitoring these tasks. A real project manager does not need any "formal" authority, he can gain an "informal" authority through his leadership and charisma.
3- The line manager has usually formal authority that is emphasized in the company's hierarchy.