Taxpayers paid for the furniture, computers and other equipment that students used at 10 Ohio charter schools. But the public doesn’t own the things it paid for, according to the state Supreme Court.

The court ruled in a split decision Tuesday that the assets are instead the property of White Hat Management, the for-profit company that once managed the 10 schools. If the schools want their chairs and desks, they’ll have to buy them back, paying for them a second time with taxpayer dollars.

That might seem incredible, but that’s what the schools agreed to in their contracts with White Hat, and the contracts have to be honored, ruled Justice Judith Ann Lanzinger, writing for the majority.

“The schools were represented by their own legal counsel, and they agreed to provisions in the contracts,” Lanzinger wrote. Unless there is fraud involved, she wrote, “ ‘courts are powerless to save a competent person from the effects of his own voluntary agreement.’ ”

Charter schools contract with for-profit management companies in many states, and the Ohio case illuminates a broader dispute over whether public money is still considered public — and therefore subject to public oversight — once it is transferred to a private entity.

In the District of Columbia, for example, the attorney general has filed two lawsuits in recent years that allege that D.C. charter school leaders used for-profit companies to divert millions of taxpayer dollars into their own pockets. The D.C. Public Charter School Board, which oversees all city charter schools, has no legal right to examine the books and records of those private companies and says that it has little ability to monitor how those dollars are spent.

In the Ohio case, the nonprofit boards responsible for the 10 schools signed contracts with for-profit White Hat and its subsidiaries in 2005. White Hat, one of the nation’s larger for-profit charter school operators, manages charter schools that serve thousands of students in three states.

White Hat ran the schools’ day-to-day operations, including hiring teachers and buying equipment for classrooms and offices. In return, the company received 95 to 96 percent of the public dollars that each school received to educate students.

Most of the schools did not fare well academically. By the 2010-2011 school year, two of the 10 schools had been closed for poor performance. The state considered one to be in “academic emergency” and four others to be on “academic watch,” according to court records.

The schools parted ways with White Hat and then sued, alleging that White Hat refused to account for how it had spent the public dollars it had received — about $90 million from 2007 to 2010, according to court documents. As part of the complaint, the schools argued that they are entitled to the physical assets, such as books and computers, that White Hat purchased with public dollars.

This week’s Ohio Supreme Court ruling means the schools are not entitled to those assets. But not every judge agreed. In a blistering dissent, Justice William M. O’Neill wrote that the majority’s opinion allowed a “fraudulent conversion of public funds into personal profit.”

The contracts between the schools and White Hat were not enforceable, he wrote, because they permitted “an operator who is providing a substandard education to squander public money and then, upon termination for poor performance, reap a bonus, paid for by public money.”