Regularity of symbolic powers of square-free monomial ideals.

We study the regularity of symbolic powers of square-free monomial ideals. We prove that if I=IΔ is the Stanley-Reisner ideal of a simplicial complex Δ, then reg(I(n))δ(n−1)+ b for all n1, where δ=limn→∞ reg(I(n))/n, b=max{reg(IΓ)|Γ is a subcomplex of Δ with F(Γ)⊆ F(Δ)}, and F(Γ) and F(Δ) are the set of facets of Γ and Δ, respectively. This bound is sharp for any n. When I=I(G) is the edge ideal of a simple graph G, we obtain a general linear upper bound reg(I(n))2n+ord-match(G)−1, where ord-match(G) is the ordered matching number of G. [ABSTRACT FROM AUTHOR]