\'Etale cohomological dimension, a conjecture of Lyubeznik and bounds for arithmetic rank

We produce a criterion for open sets in projective $n$-space over a separably closed field to have \'etale cohomological dimension bounded by $2n-3$. We use the criterion to exhibit a scheme for which \'etale cohomological dimension is smaller than what a conjecture of G.~Lyubeznik predicts; the discrepancy is of arithmetic nature. For a monomial ideal, we relate extremal graded Betti numbers and \'etale cohomological dimension of the complement of the corresponding subspace arrangement. Moreover, we derive upper bounds for its arithmetic rank in terms of invariants distilled from the lcm-lattice.