Dead persons have registered to vote, a mysterious fire has damaged the challenger's home office and the incumbent has kept her tart tongue in check successfully for another week.

With four weeks to go before the decisive Feb. 22 Democratic primary, the race for mayor of Chicago is a rich spectacle of old-fashioned American politics, where skulduggery, disaster and transformation are right at home with the trench warriors of precinct, ward and district.

Armed with a huge campaign war chest and grim determination to keep what she won in an upset four years ago, Mayor Jane Byrne holds the lead right now.

But State's Attorney Richard M. Daley, heir to the most famous City Hall name in contemporary politics, is closing the gap. Meanwhile, Rep. Harold Washington (D-Ill.) has scored well in two televised debates, with an easy manner and pulpit rhetoric aimed at bringing the minority black vote out in record numbers.

Enormous public interest has focused on the debates. The first one a week ago drew more than 2 million viewers. Part of the interest was to see whether the mayor could keep her composure and whether Daley "could speak a whole paragraph on his own," as one veteran analyst put it. Both succeeded.

The mayor, who began her political career 23 years ago as a volunteer for John F. Kennedy's presidential campaign, has portrayed herself as an experienced and successful big city executive who straightened out Chicago's rocky finances, streamlined its administration and brought new blood to the top levels of government. She never mentions Mayor Richard J. Daley, who ruled the town for 21 years until his death in 1976.

Chicagoans seem just as interested in her new calm as in her campaign claims. Her new image is chalked up to the wizardry of political adviser David H. Sawyer, who has coached Byrne into adopting a magisterial disdain in the face of her two foes.

The mayor, 48, professes surprise and some disappointment at the fascination with her new image. In a City Hall interview last week, she complained that "Everyone's talking about how I've changed. I've been wearing Pucci suits since the inaugural. That shouldn't be an issue. I'd far rather be treated as an incumbent mayor."

That's just the trouble: she is being treated as the incumbent. The dominant Chicago Tribune lamented in its Sunday lead editorial that only circumstances "impossible to imagine" could ever force it to endorse Daley--and then it endorsed him. Although admiring Byrne's spunky independence, the newspaper said her tenure has made "instability and disbelief . . . the watchwords of the city government."

As for Daley, "the best he has to offer is hope," the editorial said.

The endorsement is a gain for Daley, 40, a lawyer who served in the state Senate for eight years before defeating a Byrne-backed opponent in 1980 to become the local prosecutor.

"Richie," who will never be described as a great orator, preaches law and order, fiscal responsibility, and more jobs.

With a payroll of 40,000 people and an annual direct budget of nearly $2 billion, the stakes are high. So high that the U.S. attorney has launched a vote fraud investigation after receiving reports that substantial numbers of dead men had been registered. And high enough for someone to make the effort to torch a Daley campaign headquarters. There have been no arrests.

William J. Daley, campaign manager for his older brother, says the race will be a test of "who has the best precinct operation."

Daley has lured former vice president Walter F. Mondale to make a Feb. 6 campaign appearance with him. Meanwhile, Sen. Edward M. Kennedy (D-Mass.) has sent a letter of support for Byrne, who backed him for the Democratic presidential nomination in 1980 and again for 1984, until he backed out of that race.

Byrne campaign manager William Griffin recently pondered the situation from his camp's perspective. "Both candidates have 110 percent name recognition. The mayor's tough and the people love it."

But the Daleys, he said, "are masters at winning elections."

While the two front-runners slug it out, Rep. Washington is emerging as a strong television performer and a man willing to take a stand, however unpopular it might be. He has called for state tax increases to aid the city, and in a debate vowed to fire Police Superintendent Richard Brzeczek if he won.