On Huneke's conjecture about associated primes of local cohomology modules

A conjecture raised in 1990 by C. Huneke predicts that, for a d-dimensional Noetherian local ring R, local cohomology modules of finitely generated R-modules have finitely many associated primes. Although counterexamples do exist, the conjecture has been confirmed in several cases, for instance if $d\le 3$, and witnessed some progress in special cases for higher d. In this paper, assuming that R is a factorial domain in the main results, we establish the case $d=4$, and under certain additional conditions also the case $d=5$. Finally, when R is regular and contains a field, we apply the Hartshorne-Lichtenbaum vanishing theorem as a tool to deal with the case $d=6$.