VATICAN CITY — The preliminaries over, Catholic cardinals are ready to get down to the real business of choosing a pope. And even without a front-runner, there are indications they will go into the conclave Tuesday with a good idea of their top picks.

Then it will be just a matter of agreeing on one man to lead the church and tackle its many problems.

The conclave date was set Friday during a vote by the College of Cardinals, who have been meeting all week to discuss the church’s problems and priorities, and the qualities the successor to Pope Benedict XVI must possess.

That said, there doesn’t appear to be a front-runner, and the past week of deliberations has exposed sharp divisions among cardinals about some of the pressing problems facing the church, including governance within the Holy See itself.

The Vatican spokesman, the Rev. Federico Lombardi, said the pre-conclave meetings had given the cardinals a chance to discuss the “profile, characteristics, qualities and talents” a future pope must have.

Those closed-door deliberations, he said, provided an opportunity for discussion and information-gathering so the cardinals could go into the conclave ready to cast their ballots. “The preparation is absolutely fundamental,” Lombardi said.

Cardinal Sean O’Malley, archbishop of Boston, agreed, noting that without this week’s meetings the conclave “could drag on.”

“The preference is to have enough discussions previous so that when people go to the conclave, they already have a particular idea of who they’re going to vote for,” he told reporters at a briefing earlier this week.

Then it’s a matter of consensus-building in order to reach the two-thirds majority needed to elect a pope — a process that for the past century has taken no more than a few days.

Benedict himself was elected on the fourth round of voting in 2005, a day after the conclave began — one of the fastest papal elections in recent times. His predecessor, John Paul II, was chosen following eight ballots over three days in 1978.

In the past 100 years, no conclave has lasted longer than five days.

On Tuesday, the conclave will begin with a morning Mass in St. Peter’s Basilica, followed by a procession into the Sistine Chapel and the first round of secret balloting in the afternoon.

If black smoke is sent snaking out of the chapel chimney to indicate there is no victor, the cardinals will retire for the day. They return Wednesday for two rounds of balloting in the morning and two rounds in the afternoon, a process repeated each day, with occasional breaks for reflection, until a pope emerges.

U.S. Cardinal Timothy Dolan, considered a papal contender, said in a blog post Friday that this week’s preliminary discussions covered preaching and teaching the Catholic faith, tending to Catholic schools and hospitals, protecting families and the unborn, supporting priests “and getting more of them!”

“Those are the ‘big issues,”’ he wrote. “You may find that hard to believe, since the ‘word on the street’ is that all we talk about is corruption in the Vatican, sexual abuse, money. Do these topics come up? Yes! Do they dominate? No!”

The Americans had pressed this week for time to get to the bottom of the dysfunction and corruption in the Holy See’s governance that were exposed by the leak of papal documents last year. Vatican-based cardinals had been angling for a speedy end to the discussions, perhaps to limit the amount of dirty laundry being aired.

But by Thursday afternoon, Cardinal Roger Mahony of Los Angeles tweeted that the discussions were “reaching a conclusion” and that a mood of “excitement” was taking hold.

A Tuesday conclave start date could be read as something of a compromise. Monday had been seen as an obvious choice, to ensure a pope would be elected and installed well ahead of the busy Holy Week that precedes Easter, beginning with Palm Sunday on March 24.

According to Vatican analysts, the list of papabili, or those considered to have the stuff to be pope, remains relatively unchanged since the 85-year-old Benedict first announced he would resign on Feb. 28, kick-starting the papal transition.

But some Italian media have speculated that with governance such a key issue, the cardinals might also be considering an informal pope-Vatican secretary of state “ticket.” The secretary of state — who is primarily responsible for running the Holy See — is not an elected position but a papal appointment.

Also Friday, the cardinals formally agreed to exempt two of their voting-age colleagues from the conclave: Cardinal Julius Darmaatjadja, the emeritus archbishop of Jakarta, who is ill, and Scottish Cardinal Keith O’Brien, who resigned last week after admitting to sexual misconduct.

That formality brings the number of cardinal electors to 115, two-thirds of whom — or 77 — must vote for the victor. Benedict in 2007 changed the conclave rules to keep the two-thirds requirement; Pope John Paul II had decreed that only a simple majority would be needed following 12 days of inconclusive balloting.

By reverting back to a two-thirds vote, Benedict was apparently aiming to ensure a consensus candidate emerges quickly, and to rule out the possibility that cardinals might hold out until the simple majority kicks in to push through their candidate. The decision may prove prescient, given the apparent lack of a front-runner in this conclave.

Lombardi said a few items of business remain outstanding, including drawing lots for rooms at the Vatican’s Santa Marta hotel, where the cardinals will be sequestered once the conclave begins.

On Friday, he showed a video of the room where the new pope will spend his first night as pontiff: It features a bed with a dark wood headboard and a carved image of Christ’s face, as well as a sitting area and a study.

The pope is expected to stay there for a few weeks until the papal apartment in the Apostolic Palace can be renovated. The apartment was sealed Feb. 28, just after Benedict resigned, and cannot be reopened until the new pope formally takes possession.