Desingularization by char (X) -Alterations

This is an extended abstract of my talk at the “Workshop on Positivity and Valuations” at Centre de Recerca Matemàtica. The talk was devoted to my recent work (Temkin, Tame distillation and desingularization by p-alterations, [8]), in which I prove that any qe integral Noetherian scheme X can be desingularized by an alteration X^′→ X whose degree [ k(X^′) : k(X) ] is only divisible by primes non-invertible on X. I had already reported on this work in the Oberwolfach Research Institute for Mathematics, and the first two sections of this abstract are close to my Oberwolfach report. In addition, I added a new third section where further conjectures and directions of research are discussed.