Senate Majority Leader Mitch McConnell (R-Ky.) gave a knowing smile when the only hiccup of Thursday’s formalities got resolved: After a brief pause, Chief Justice John G. Roberts Jr. realized he had to gavel the impeachment trial out of session, adjourning until Tuesday.

McConnell got the day’s ending he had hoped for, a Senate looking somber and serious as it conducts President Trump’s impeachment trial. A Senate that might, as he put it a day before, be capable of rising above “short-termism and factional fever.”

But that fever lingers as the sides jockey for position when the real phase of the trial kicks in, with early skirmishes about calling witnesses getting overtaken by talks about what limits could be placed on the presentation of evidence.

And the biggest issue, some senators say, could come when Trump’s legal team takes its allotted time to rebut the case from the House impeachment managers.

So far, more than four months after the Ukraine scandal broke, Trump and his advisers have given sparse explanations for their side of the story. The White House press office quit holding a daily briefing many months ago, leaving public comments to those moments when Trump engages in impromptu gaggles with the media.

His lawyers sent angry letters to House Democrats during their impeachment proceedings that denounced what Trump considered “profound procedural deficiencies” but did not spell out the facts of the case. The president declined to send his legal team to the House Judiciary Committee’s hearings, instead leaving his GOP allies to make arguments on his behalf.

McConnell, knowing that he currently has the votes to acquit Trump, does not want the Senate trial to come off looking anything like the partisan brawl in the House or a typical Trumpian event. With a handful of GOP incumbents facing difficult reelections in November, McConnell wants a quick, clean, dignified trial to protect their political fortunes as well as his own.

That’s why some Democrats believe his defense team could provide the most dramatic moment of the trial.

“I think they have to make a choice when the lawyers show up to the Senate, so that’s kind of a fulcrum moment,” Sen. Chris Murphy (D-Conn.) said in an interview, explaining the Democratic anticipation for the Trump team’s presentation. “Do they put on evidence of massive Ukrainian corruption and why President Trump withheld aid until that corruption was cleaned, or do they claim that it is all hearsay?”

Some Republicans share that anticipation for the Trump lawyers, believing the moment will finally give them the opportunity to forcefully defend the president.

“He hasn’t had a chance to make his case. He’ll have that chance in the Senate. The Senate owes that to him, and I think the Congress owes that to the country,” Sen. Roy Blunt (R-Mo.) told reporters Tuesday.

And some Republicans who have been open to calling more witnesses said their decision could hinge on the president’s lawyers.

“You know I haven’t heard from the president’s team, his defense. So I’d like to hear the president’s defense,” said Sen. Mitt Romney (R-Utah).

But no one knows for sure what approach the team will take and whether it will prompt a series of objections from the House managers or the Democratic jurors over the nature of the evidence they present.

So far Trump’s senior advisers have only signaled that they will not likely take three days to present their case, as former president Bill Clinton’s lawyers did during the last impeachment trial in 1999, and believe the trial could wrap up in about two weeks. At a briefing for reporters Wednesday, presidential officials suggested they would be fine without calling additional witnesses — but where they go with their own presentation is unclear.

And Trump’s aides have often said one thing only to be preempted by the president’s mercurial decisions, leading some to wonder what orders he will give his lawyers when it is finally their turn.

Sen. Richard Blumenthal (D-Conn.), as he entered Thursday’s preliminary proceedings, said he is still trying to learn how Democrats can object if Trump’s lawyers wander into territory that they believe is unrelated to the case: “What’s the procedure for blocking that evidence? Do the managers have to object? Can senators object? And how is it going to be done?”

They have argued against the testimony of Hunter Biden, son of former vice president Joe Biden, whose former position on the board of a Ukrainian gas company has been at the center of Trump’s effort to pressure that nation’s leaders into investigating the Bidens.

But few senators seem to grasp what rules govern evidence, for either the House managers or Trump’s lawyers.

“I think there’s not a limitation on what they can present,” Sen. Rand Paul (R-Ky.) said Thursday, suggesting both sides should have a pretty free hand in presenting whatever case they want to make.

If the president’s legal team wants to diverge deeply into Hunter Biden, that is fine, Paul said. “How you defend yourself would be up to your team.”

Other Republicans, upon a steady stream of new documents coming from House Democrats, argued that House Intelligence Committee Chairman Adam B. Schiff (D-Calif.), the lead manager, should be limited only to the evidence that was presented up to the Dec. 18 votes on two articles of impeachment for abuse of power and obstruction of Congress.

The Senate’s official rules on impeachment do not give much specificity on the matter. “The presiding officer on the trial may rule on all questions of evidence including, but not limited to, questions of relevancy, materiality, and redundancy of evidence and incidental questions,” Rule VII states.

That suggests that senators can voice objections to Roberts, whose options would be to make a ruling and see if senators call for a vote on that ruling, or just put the objection up to a vote of the full Senate

This runs the risk of turning the trial into a series of stops and starts, as Republicans, or Trump’s team, object to presentations by the Democratic managers, or the managers object to the president’s opinions.

All that would destroy McConnell’s vision of a Senate rising above “factional fever,” resembling the battle that took place in the House.