  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[x_0,..., x_d], 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+1. This indicates that the tight closure of an ideal
behaves more uniformly than the ideal itself. Moreover, if R is normal, one
obtains a generic bound for membership in the Frobenius closure. If d is at
most 2, then the bound for ideal membership in P can be computed from the known
cases of the Froeberg conjecture and yields explicit generic tight closure
bounds.
