MOSCOW — Heimir Hallgrimsson wanted to save time. Before Iceland’s coach faced the questions of the gathered news media on the eve of his country’s first game at a World Cup, he tried to help out. “Before anyone asks,” he said, “I’m still a dentist, and I will never stop being a dentist.”

Hallgrimsson and his players are in no doubt as to how they are perceived; they are well aware of the role to which they have been assigned. They are the ultimate underdogs, the smallest nation ever to play in a World Cup — just as, two years ago, they were the smallest nation ever to play at a European Championship — the team that knows a substantial proportion of its fans by name, the team that is managed by a dentist.

They understand it, too. They appreciate just how compelling their story is, how remarkable it is that a country so small should now have assumed a place if not in the first rank of soccer nations, then surprisingly close to it. The romance of their rise is so seductive that the players cannot help but acknowledge it and, occasionally, even revel in it.

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There comes a point, though, where that romance begins to obscure the achievements of these players, rather than celebrate them, where the appeal of presenting them as nothing but a plucky band of adventurers does not highlight the scale of what they have achieved, but begins to diminish it.