The Constitution has already been cited look in the thread.The Constitution requires a majority vote and the Supreme Court upheld that. Pretty simple concept.
Feel free to quote article, section and clause that says a majority vote is required for a State to secede.
None of which specifically regards secession. The OP is correct, it all boils down to the 10th Amendment.
Amendment X
The powers not delegated to the United States by the Constitution, nor prohibited by it to the States, are reserved to the States respectively, or to the people.
Since the subject of secession in NOT ADDRESSED as a federal power, it is left to the States because it is not prohibited by it to the States either.