London — The core of the crisis approaches. The longer the Falkland Islands remain in Argentine hands, the more difficult the British government's strategic choices become.
So far, Prime Minister Margaret Thatcher has widespread support in her own party, in the House of Commons, among the British public, in the European Community, in the United Nations Security Council, and in the United States for her stick-and-carrot campaign to regain control of the remote islands and their 1,800 people. Her naval task force, steaming closer and closer to the islands, adds muscle to her diplomacy.
The crunch will come if Argentina refuses to be intimidated, and the outline of an Argentine naval ship appears in the periscope of a British submarine inside the 200-mile exclusion zone.
Mrs. Thatcher has said that such an appearance would mean that Argentina had abandoned the search for a peaceful solution. But if she orders shots to be fired, she could find her own position greatly weakened.
Public opinion here, in the United States, and in the UN could swing against her. The NATO alliance could be strained. The Russians could benefit.
The US would be under strong pressure to take open sides with London, strengthening NATO but weakening the American posture in Latin America even further.
Here at home, government ministers have already said they are looking at the possibility of raising taxes to pay for a lengthy naval stay in the South Atlantic. The pound sterling and London share prices have both fallen just at a time when inflation and interest rates were also coming down.
For the moment London still puts diplomacy first and force second.
By sending Foreign Secretary Francis Pym to Washington to talk to Secretary of State Alexander M. Haig Jr., Mrs. Thatcher has kept alive the Haig mediation process, even though she and her Cabinet are cool, to say the least, to the Argentine proposals relayed by Mr. Haig late April 19.
She agrees with Winston Churchill's remark: ''To jaw-jaw is better than war-war.''
Prominent newspapers such as The Times of London, and Conservative backbenchers, thunder that Britain is defending universal principle, and that if Britain does not stand firm against Argentina, the danger to Belize from Guatemala, to Guyana from Venezuela, to Chile from Peru and Bolivia -- even one day to the US Southwest from Mexico -- would be heightened.
Public opinion polls give strong support to Mrs. Thatcher's use of the task force as a diplomatic stratagem. The Security Council voted 10 to 1, with the Soviet Union abstaining, for an immediate Argentine withdrawal and talks.
In a series of well-publicized, televised moves, Britain has brandished its military stick by taking over 35 civilian ships, including the passenger liners Canberra and Uganda, three oceangoing tugs, and a variety of ferries, to carry extra troops, aircraft, and medical services to the task force.
Aging Vulcan bombers flash across television screens as they are converted from nuclear to conventional weapons, signaling Buenos Aires that they might be used to ''crater'' Argentine airfields to establish an air as well as naval blockade around the islands.
British correspondents with the task force are permitted to report only what the British government wants Argentina to know. London regulates the reporting, in a way American governments could not, by issuing so-called ''D'' notices, which forbid mention of information including location and size of the force, or possible engine or other difficulties.
Meanwhile, as Mr. Pym carried new proposals to Mr. Haig and President Reagan, Mrs. Thatcher was insisting on two points:
1. The islanders themselves must be allowed to determine their own future and must have a period in which to recover from and think about the invasion once the Argentines leave.
2. Argentina must withdraw completely and haul down its blue and white flag before Britain will negotiate the future of the islands.
Argentina had apparently suggested joint administration, a withdrawal of its own forces, and the turning-back of the British task force.
As a carrot, however, the British have not ruled out anything in talks to come. It is considered virtually certain that London will end up by yielding sovereignty in one way or another, while trying to retain the islanders' right to keep living in a democratic, British way.
Mrs. Thatcher's difficulties now intensify.
If she establishes a staging point on South Georgia, 800 miles east-southeast of the Falklands, or if she announces an air blockade of the Falklands, she will be supported here.
But if British marines, pilots, or islanders themselves are killed, worried voices will grow louder here at home, and Mrs. Thatcher could well be faced with the loss of vital political support.
Labour and Liberal Party spokesmen have already said in recent days that diplomacy must continue at all costs. Even some Tory backbenchers stress the need to avoid shooting.
Denis Healey, Labour spokesman on foreign affairs, advocates that UN Secretary-General Javier Perez de Cuellar should take over the mediator's role, so that the US can impose economic sanctions against Argentina and persuade Japan to do the same.
Government sources close to Mrs. Thatcher commented April 21 that they believe the US is ''instinctively, spiritually, and morally'' on the side of the democracy in the dispute.
British public opinion has supported the task force so far, but more than 50 percent in one poll felt the islands were not worth losing lives for.
It is feared here that some American opinion, mindful of possible Soviet advantage, might turn against Britain in a shooting war. Latin American and UN General Assembly opinion would also be opposed.
Mrs. Thatcher would retain her die-hard backbench support, but if Argentina still refused to give in, she would be in danger of losing her job altogether, and following former Foreign Secretary Lord Carrington into retirement.