ALL eyes turn to Jesse Jackson.

Washington Mayor Marion S. Barry Jr. is arrested on drug charges. Barry is checked into a rehabilitation program. Washington's mayoral race is thrown into a tailspin.

All eyes turn to Jackson. Will Jesse run?

Jackson kindled speculation that he would challenge Barry when he changed his official residence to D.C. last July, only weeks before the qualification deadline for mayoral candidates.

But Jackson said he would not run. His ties with the D.C. mayor run deep, and even Barry's unrelieved record of scandal and corruption evidently could not persuade Jackson to cross his old friend.

Then Barry was arrested.

Many believe that Jackson should reconsider his earlier proclamation. He is, after all, Barry's logical successor. And if elected D.C. mayor, he could finally overcome a perennial criticism of his presidential bids--that he has no experience in elected office.

Many of Jackson's supporters would be delighted to see him run for mayor for this reason. But the people most excited about the prospect of Mayor Jackson are Democrats who want to get rid of Jackson.

Washington is a hellish city to govern. Guns, crack and poverty prevail. The city bureacracy is a quagmire. The district does not even have final authority over its own budget. The Washington Monthly, a national political journal, recently proclaimed the D.C. administration "the worst city government in America."

Democratic beacons assume that the troubles in Washington are so out of hand that any Washington mayor will come out looking like a failure. As D.C. mayor, Jackson would be shunted out of the national limelight and out of contention for the Democratic presidential nomination.

These Democrats secretly hope that Barry's arrest will rid them once and for all of two embarrassing problems--Barry and Jackson.

In 1992, they hope, the Jackson problem will be as good as gone.

THE "Jackson problem" was the dilemma faced by Democrats in 1988 when Jackson's constant presence and unexpected electoral success threatened Gov. Michael S. Dukakis' claim to the leadership of the party. It was the fear that alienating Jackson would alienate Black voters, the most loyal voting bloc in the Democratic coalition.

The "Jackson problem" was the public fear that Jackson would press on the temple walls and bring the party down around him.

Thus, those looking to 1992 want Jackson out of the picture, permanently.