All four Democratic contenders agree on how to increase education spending: by taxing the booming natural gas industry. They argue that Pennsylvania is giving away the store to drillers in the Marcellus shale formation by not collecting more.

Allyson Y. Schwartz, a Democratic congresswoman from the Philadelphia suburbs running for governor, called on the state to “demand they give us 5 percent and use the money to invest in education.”

The governor opposes adding new taxes to fees that drillers already pay, mostly to the counties where they operate. The argument the other side makes, he said, is that the gas is the property of all Pennsylvanians.

“No, it’s not,” he said. “It’s the property owner’s gas. I’m sorry: It’s the mineral owner’s gas.”

His invoking of property rights was a reminder that the governor hews strongly to conservative principles, even in a state that has a moderate tradition. Elected in the Tea Party wave of 2010, Mr. Corbett has refused to expand Medicaid under the president’s health care law, and he has presided over corporate tax cuts of $1.2 billion.

“You can go all the way back to Reagan: Reduce your taxes, you’re going to see growth,” he said.

But it is unclear whether the economy will benefit the governor. Pennsylvania’s 6 percent unemployment rate has dropped 2.1 points during his term. But the state ranked 48th in job creation last year, according to the federal Bureau of Labor Statistics. Standard & Poor’s recently scolded state lawmakers for a budget out of whack between revenue and spending. In a year when other states have surpluses, Pennsylvania is facing a $1 billion deficit, for which critics say the governor’s business tax breaks are to blame.

“Tom Corbett has dug us into a hole because he has been ideologically driven to say no new revenues no matter what,” Ms. Schwartz said in the debate.

With the Democrats agreeing on their anti-Corbett line, they have turned on one another ahead of their primary. Mr. Wolf, a businessman from York, spent nearly $4.5 million of a personal fortune in an early airwaves barrage, portraying himself as a folksy outsider concerned about his employees. He opened a double-digit lead.