We study a singularity of a pair, consisting of a smooth variety and a multi-ideal, defined over a field of positive characteristic. Singularities of a variety defined over a field of characteristic 0 are extensively studied by making use of many good properties which are not available for the positive characteristic case. So we cannot expect the same results in positive characteristic by the same method as in characterisitc 0. In my talk I will propose another way to prove the results which are known in characteristic 0. The way is to construct a bridge between positive characteristic and characteristic 0 by means of inversion of modulo p-reduction and discussion of arc spaces. I will introduce our expected conjecture and its consequences. Then, I will show my recent results towards the conjecture. Actually, for every ‘multi-ideal’ on a smooth variety over the base field of positive characteristic, there exists its lifting ‘multi-fractional ideal’ on a smooth variety over characteristic 0 with the same mld (minimal log discrepancy). By this bridge, we can induce some results on mld for positive characteristic from those for characteristic 0.
This video was produced by the Japan-US Mathematics Institute and forms part of JAMI Conference 2022.
