Splittings and symbolic powers of square-free monomial Ideals

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.