Embedded associated primes of powers of square-free monomial ideals

An ideal II in a Noetherian ring RR is normally torsion-free if Ass(R/It)=Ass(R/I)Ass(R/It)=Ass(R/I) for all t≥1t≥1. We develop a technique to inductively study normally torsion-free square-free monomial ideals. In particular, we show that if a square-free monomial ideal II is minimally not normally torsion-free then the least power tt such that ItIt has embedded primes is bigger than β1β1, where β1β1 is the monomial grade of II, which is equal to the matching number of the hypergraph H(I)H(I) associated to II. If, in addition, II fails to have the packing property, then embedded primes of ItIt do occur when t=β1+1t=β1+1. As an application, we investigate how these results relate to a conjecture of Conforti and Cornuéjols.