Generic Bounds for Frobenius Closure and Tight Closure

We use geometric and cohomological methods to show that given a degree bound for membership in ideals of a fixed degree type in the polynomial ring P = k[x1, . . . , xd], one obtains a good generic degree bound for membership in the tight closure of an ideal of that degree type in any standard-graded k-algebra R of dimension d. This indicates that the tight closure of an ideal behaves more uniformly than the ideal itself. Moreover, if R is normal, one obtains a general bound for membership in the Frobenius closure. If the dimension d ≤ 3, then the bound for ideal membership in P can be computed from the known cases of the Fröberg conjecture and yields explicit generic tight closure bounds.