  We study the symbolic powers of square-free monomial ideals via symbolic Rees
algebras and methods in prime characteristic. In particular, we prove that the
symbolic Rees algebra and the symbolic associated graded algebra are split with
respect to a morphism which resembles the Frobenius map and that exists in all
characteristics. Using these methods, we recover a result by Hoa and Trung
which states that the normalized $a$-invariants and the Castelnuovo-Mumford
regularity of the symbolic powers converge. In addition, we give a sufficient
condition for the equality of the ordinary and symbolic powers of this family
of ideals, and relate it to Conforti-Cornu\'ejols conjecture. Finally, we
interpret this condition in the context of linear optimization.
