Mayor Jane Byrne will go "house hunting" Tuesday in this city's infamous Cabrini-Green public housing development, and says she may rotate her residence among other Chicago housing projects in an effort to combat rising crime.

In a news conference today in her fashionable high-rise apartment one block from the historic Water Tower on North Michigan Avenue and six blocks from Cabrini-Green, Byrne said she expects to buy some furniture and get resettled within two weeks.

"Some of you may find it hard to believe, but I've lived in public housing before," the mayor said, referring to the years of her first marriage to a Navy flier who later died in an air crash. "It doesn't come as a big shock to me."

But shock was only one of the words used by Byrne's friends and political foes. Typical was the remark of Renault Robinson, a black member of the Chicago Housing Authority board and a frequent Byrne critic. "I'm happy to hear that a white female mayor would have the courage to move into Cabrini-Green," he said. "Frankly, that takes guts."

The Cabrini-Green community of 14,000 persons in 81 high-rise buildings and row houses, long a symbol of urban neglect, has been the scene of 10 murders in the last nine weeks. Police say 35 other persons have been wounded by random sniper attacks, and the area has been the scene of warfare between gangs fighting for control of narcotics and extortion rackets in the 70-acre project.

Fear has so penetrated the project that at times police and city ambulance units have refused to aid residents in the upper floors of high rises.

Byrne's decision late Saturday to move into the low-income project to dramatize the public housing crisis came after she had made daily inspections of the neighborhood during the last two weeks. She was especially upset Saturday when she encountered a young woman who a few minutes earlier had been raped in the street.

The mayor said today she was convinced that police service had deteriorated in the neighborhood and that her presence would reverse that pattern.

"Wherever a mayor goes there seem to be city services galore," Byrne said. "The mayor of the city of Chicago has a lot of power, and it's time for Cabrini-Green to get that power personally because this is a cancer that could spread into every neighborhood in Chicago."

Byrne said her normal complement of security personnel would accompany her when she moves into a CHA building. She also said she is considering establishing multiple residences in other CHA projects around the city so that gang members and city workers would not know precisely where she was staying. This, she said, would improve services and deter violence.

The mayor insisted that her gesture is more than symbolic, and indicated today she was willing to stay in a CHA apartment for six months or more, "for as long as it takes to tell the community they are going to live without fear. I want to show people they don't have to be afraid."

And she scoffed at suggestions that her move was designed to better failing public relations. "Whoever says that should take the next apartment," Byrne said.

Cabrini-Green residents had mixed reactions to their prospective new neighbor. Some tenants praised the idea, but others said it would do little good. "What's going to happen is going to happen anyway while she's asleep," one member of a tenant council said.

Byrne's action is not unprecedented. Atlanta Mayor Maynard Jackson spent two nights in Atlanta's Bankhead Court in October 1974 after incidents of rock-throwing and other violence plagued the project.