  In this article, we study certain local cohomology modules over $F$-pure
rings. We give sufficient conditions for the vanishing of some Lyubeznik
numbers, derive a formula for computing these invariants when the $F$-pure ring
is standard graded and, by its means, we provide some new examples of Lyubeznik
tables. We study associated primes of certain Ext-modules, showing that they
are all compatible ideals. Finally, we focus on properties that Lyubeznik
numbers detect over a globally $F$-split projective variety.
