"No part of your argument has to do with racial or ethnic profiling?" asked Chief Justice Roberts, and the SG agreed.

The Arizona immigration case is emphatically not about the issue that seems to be all people opposed to the law argue about in the political sphere.

As in the Affordable Care Act case, it was Solicitor General Donald B. Verrilli and former Solicitor General Paul D. Clement arguing on opposite sides. Unlike in that case and because the law under attack is a state law, Verrilli is arguing against the choice of the democratic majority, and Clement is arguing to uphold it.

Mr. Clements [sic] said the state was making an effort to address an emergency situation with a law that complemented federal immigration policy. “Arizona borrowed the federal standards as its own,” he said.

Mr. Verrilli countered that Arizona’s approach was in conflict with the federal efforts. “The Constitution vests executive authority over immigration with the national government,” he said.

That is, it's a federalism case, not a constitutional rights case. The question is the allocation of power between the federal and state government. (The Affordable Care Act case is also a federalism, and not a rights case, even though in the political sphere, people opposed to the individual mandate concentrate on the imposition on the individual, not misallocation of power as between the federal and state governments.)

“What does sovereignty mean if it does not include the ability to defend your borders?” Justice Antonin Scalia asked.

Chief Justice Roberts said the state law merely requires that the federal government be informed of immigration violations and leaves enforcement decisions to it. “It seems to me that the federal government just doesn’t want to know who is here illegally and who’s not,” he said....

Chief Justice Roberts, writing for four of the justices in the majority, said the state law under review “simply seeks to enforce” a federal ban on hiring illegal workers. “Arizona went the extra mile,” he wrote last year, “in ensuring that its law closely tracks” the federal one.

That was from the Adam Liptak article in the NYT. Here's Lyle Denniston at SCOTUSblog:

In an oral argument that ran 20 minutes beyond the scheduled hour, the Justices focused tightly on the actual operation of the four specific provisions of the law at issue, and most of the Court seemed prepared to accept that Arizona police would act in measured ways as they arrest and detain individuals they think might be in the U.S. illegally.  And most of the Justices seemed somewhat skeptical that the federal government would have to change its own immigration priorities just because states were becoming more active....

The Court’s three more liberal Justices — Stephen G. Breyer, Ruth Bader Ginsburg and Sonia Sotomayor — offered what appeared to be a less than enthusiastic support for the federal government’s challenge....

Kagan has recused herself, by the way. It seems rather obvious that Arizona will win this case. The first quote in Liptak's article is Sotomayor saying to Verrilli: "You can see it’s not selling very well."