An important problem in commutative algebra is studying the relationship between symbolic and ordinary ideals. One striking result in this direction was found by Ein-Lazarsfeld-Smith, who showed that for regular rings in characteristic 0, the dn-th symbolic power of any ideal is contained in the n-th ordinary power of that ideal, where d is the dimension of the ring. Their method proved to be quite powerful, and was adapted to the positive characteristic setting by Hara and the mixed characteristic setting by Ma and Schwede.
In this talk, we will discuss an approach to extending Ein-Lazarsfeld-Smith’s result to the non-regular setting by coming up with a new subadditivity formula for test ideals. Recent joint work with Javier Carvajal-Rojas shows that this approach works for segre products of polynomial rings. Afterwards, we’ll talk about how applying this approach to any toric variety reduces to solving a certain combinatorial problem.