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But after the Supreme Court's Grutter decision upheld the use of race in holistic admissions plans, UT-Austin restored a racial component to its program. Race was added as a factor to UT-Austin's "personal achievement index," a mix of leadership qualities, extracurricular activities, work and service experience, and special circumstances. That index, known as the PAI, and a separate academic index are used on a matrix to grant admission to applicants who don't get in through the Top Ten Percent law.

No goals or racial quotas are established. But, while a federal judge described the university's use of race as "a factor of a factor of a factor," all sides acknowledge that in an individual case, race can be the determining factor. (The university insists that Ms. Fisher would not have been admitted even if she had received the highest PAI score.)

In its Supreme Court brief, UT says its selectivity is hindered by the Top Ten Percent plan, and its use of race is designed to bolster racial diversity in a broad sense, including within minority groups, such as by attracting better-credentialed black and Hispanic students than under the automatic plan.

Black and Hispanic students admitted through the holistic program, such as "the African-American or Hispanic child of successful professionals in Dallas," have "great potential for serving as a 'bridge' in promoting cross-racial understanding, as well as in breaking down racial stereotypes," the university's brief says.

While Ms. Fisher has a relative handful of groups, mainly conservative-leaning, on her side, the university has attracted some 70 friend-of-the-court briefs. Its supporters include President Barack Obama's administration, most higher education groups and many individual institutions, retired military leaders, Fortune 100 corporations, and K-12 groups.

"In higher education, we don't sense any lessened need for the right to appropriately consider race and ethnicity in considering the makeup of our student bodies," said Ada Meloy, the general counsel of the Washington-based American Council on Education, the main umbrella group for colleges and universities. "To change the rules now would be highly disruptive to higher education."

Changes on the Bench

The case will be decided by just eight members of the Supreme Court. Justice Elena Kagan is recused because she had some involvement in the litigation as the U.S. solicitor general early in the Obama administration.

More significantly, the author of the Grutter opinion, Justice O'Connor, is retired, and her replacement, Justice Samuel A. Alito Jr., is much more skeptical of race-based government policies. So is Chief Justice John G. Roberts Jr., who was not part of the Grutter court but wrote the court's opinion in the 2007 case of Parents Involved in Community Schools v. Seattle School District, which sharply limited the ways K-12 schools may consider race.

"These shifts in personnel sometimes have enormous consequences," said Pamela Harris, a visiting professor at Georgetown University Law Center, in Washington, and a former director of its Supreme Court Institute. "I think this is an issue the chief justice really and desperately cares about."

"I think he would prefer to go broad," with a decision barring consideration of race, she added.

But the member of the court to watch, observers agree, is Justice Anthony M. Kennedy, whose vote would be needed to limit or strike down race in admissions given the likely split among the court's other members. Because the federal 5th Circuit appeals court upheld UT's plan, a 4-4 division in the Supreme Court would affirm that decision without handing down an opinion or setting a precedent.

Justice Kennedy dissented in the Grutter case, and joined the majority in the Parents Involved case, but wrote a crucial concurrence that kept open the possibility of some use of race. He has generally endorsed the idea expressed by the late Justice Lewis F. Powell Jr. in the 1978 Regents of the University of California v. Bakke case that racial diversity can further an educational institution's mission.

But Justice Kennedy has always called for weighing such programs under the most exacting of constitutional scrutiny, and he has never actually voted to uphold a race-conscious plan.

Most of the briefs aim their arguments squarely at Justice Kennedy, with UT and Ms. Fisher's side mentioning him 20 times each.

"We know from Justice Kennedy's inquiries in the [Parents Involved] case that he will wonder why" a race-neutral alternative like the Top Ten Percent plan is not enough to satisfy the university's diversity needs, said Mr. Negron of the NSBA. "We know that's the kind of question he will be asking."