"Don't forget, you are my daughter," scrawled Lucille Ball on the sheet music for They're Playing Our Song. And though Lucie Désirée Arnaz is headed east to Broadway in that new show—a Neil Simon-Marvin Hamlisch-Carole Bayer Sager collaboration—there's a fat chance that she will forget. "When people see me, they see a history of my mother," explains Lucie, 27. "They say, 'I didn't know you could dance—your mother can't dance.' I want to say, 'Hey, what about my father?' But I'm used to it. I could be compared to worse people." Indeed. Lucie's odds-on to become another Hollywood legacy like that superstar of the stage (who once lived with Lucie's brother, Desi Jr.), Liza Minnelli.

When They're Playing Our Song debuted at the L.A. Music Center last month, Lucille, 67, told her daughter: "I envy you. It looks like you're going to be stuck in a hit for a year." "Which means," translates young Lucie, "I'll be in New York for a year and it's painful for her." She values her mom's tips on playing the part, but shrugs: "A mother watching a daughter is not objective. I have 20 laughs to [co-star] Robert Klein's 90, and that's not enough for Mother. I don't measure a character by laughs," continues Lucie. "I tell her, 'You made history by never delivering one-liners. Your fame came from being a bizarre person.' "

That flibbertigibbet TV persona aside, Lucille was a strict, attentive and loving mother who was 39 when Lucie was born. "She waited 10 years and had two miscarriages, so it was good news that at least she could get pregnant," says Lucie. "I don't think I could try 10 years—after the fifth year, I'd adopt. When we were growing up," Lucie continues fondly, "she was there to say goodbye in the morning and always home for dinner."

Her parents were divorced in 1960 when Lucie was 8 and Desi was 7. "I felt sorry for my father, the one that had to leave," she recalls. Always close to her Cuban-born bandleader dad, it took awhile to adjust to her stepfather, Gary Morton. "He didn't try to take Dad's place," she says appreciatively. "He was our mother's husband. Both Mother and Dad have tempers, which is one of the reasons they didn't survive together. They should have been given the Nobel Peace Prize for lasting 20 years." Though she has her mother's legs and her father's Latin darkness, Lucie doesn't really look like either parent and cracks, "I've always wanted to take a good long look at the mailman."

When Lucie was 8, Mother converted their garage into a theater as a Christmas present. "It had a little yellow curtain, one pink light and rented chairs," says Lucie, who with friends charged five cents a seat for a lip-synched production of Bye Bye Birdie. "It was just a girls' group. We were way ahead of the times," she laughs. Weekends were often spent with her father. "Mother hated that," Lucie remembers. "She would say, 'I always get you when I have to yell, "Did you do your homework?" He gets you for the fun stuff.' " Always frugal, Lucille gave her daughter a 50-cents-a-week allowance and still watches her spending. (When she heard the rent on Lucie's first apartment had shot up, Mom directed, "You have to move. You can't afford that.")

In 1961 Lucie moved to New York with her mother, who was starring on Broadway in Wildcat. She sat through it 17 times and decided movies couldn't compare with live theater. But, at 16, she joined her mother for six years on the Here's Lucy TV show. There, she met Phil Vandervort, an actor-producer, whom she married on her 20th birthday.

"Mother couldn't object," she figured, "because there were a lot of others into drugs and I chose this pulled-together, little-bit-older [nine years], creative, good-looking guy. She was afraid if she complained, I'd go out with some lunatic." Phil considered Lucie too young but, she says, "I wouldn't let him leave me. I wanted to be a married lady. It lasted less than a year—and I got it out of my system."

Since Phil, Lucie has lived alone. "But I've gotten to know a lot of different kitchens," she adds. "I've been infatuated with 40 men, but have only gone bananas over three people in my life. Yet I'm in love with love and am thrilled to have a show that will tie me down and keep me out of trouble." Having a mother "who did it all," she too wants children, but says, "It is hard to be pregnant and do a Broadway show." She owns a two-bedroom bungalow with pool in L.A. and a loft in New York's Chelsea. "I enjoy the energy of Manhattan and the quiet of L.A. I'm truly," she quips, "bi-coastal."