Associated primes and primary decompositions of Frobenius powers

By the work of Smith-Swanson and of Sharp-Nossem, the linear growth property of primary decompositions of Frobenius powers of ideals in rings of prime characteristic has strong connections to the localization problem in tight closure theory. The localization problem has recently been settled in the negative by Brenner and Monsky, but the linear growth question is still open. We study growth of primary decompositions of Frobenius powers of dimension one homogeneous ideals in graded rings. If the grading is positive over a field we prove that the linear growth property holds. For non-negatively graded rings we are able to establish polynomial growth. We present explicit primary decompositions of Frobenius powers of an ideal, which were known to have infinitely many associated primes, having this linear growth property. We also study the set of associated primes of Frobenius powers of the ideal in the example of Brenner and Monsky. We prove that this set is infinite. We discuss how this result relates to the linear growth question, and to the localization problem as well.