Findlaw has an overview of the relevant Supreme Court cases for education deriving from the 14th Amendment. One past decision that caught my eye was the Swann (1971) case, described as follows:
Because current attendance patterns may be attributable to past discriminatory actions in site selection and location of school buildings, the Court in Swann determined that it is permissible, and may be required, to resort to altering of attendance boundaries and grouping or pairing schools in noncontiguous fashion in order to promote desegregation and undo past official action; in this remedial process, conscious assignment of students and drawing of boundaries on the basis of race is permissible. Transportation of students--busing--is a permissible tool of educational and desegregation policy, inasmuch as a neighborhood attendance policy may be inadequate due to past discrimination. The soundness of any busing plan must be weighed on the basis of many factors, including the age of the students; when the time or distance of travel is so great as to risk the health of children or significantly impinge on the educational process, the weight shifts.The Court here was analyzing remedies for past discrimination by school district officials. But what if there has been no past discrimination in the location, quality, and attendance of public schools?
From the descriptions of the case and my quick skimming of the arguments, it seems that some students are denied opportunities available to others solely based on their race. I do not see how this can be reconciled with the Equal Protection Clause as it has developed through the two University of Michigan cases. In the undergraduate case, a majority of the Court ruled that the admissions system (a point system that awarded extra points for members of minority groups) was too "mechanistic." It seems that these two systems are even more mechanistic. I'm guessing this will be 5-4 to strike down both systems.
So what are school systems supposed to do to ensure Equal Protection? In a degenerate way, having only one school in each district would do it, as would having completely random assignments if there are multiple schools. Beyond that, we could start with a system of schools constructed and staffed as similarly as possible and allow choice, requiring that every school accept all applicants. As long as those two requirements were maintained--identical offerings and a requirement to take all applicants--then there should be the presumption that everyone has the same opportunities. But this would do nothing to ensure "diversity" as it is being used in this context.