Birational Invariants and Decomposition of the Diagonal

Variation of stable birational types in positive characteristic

“…The purpose of this section is to prove the following theorem, which shows that decompositions of the diagonal allow to distinguish two stable birational types in much greater generality. Partial results in this direction have previously been obtained by Voisin [Voi19].…”

“…If π is smooth, then the above theorem is known and follows by similar arguments as in [Voi19]. However, for most applications (e.g.…”

“…Motivated by a recent result of Shinder [Shi19] with an appendix by Voisin [Voi19], we prove in this paper that the non-existence of a decomposition of the diagonal does not only prevent the variety from being stably rational, but in fact from being stably birational to a very general hypersurface of any given degree and dimension.…”

“…Since [NS19], as well as Kontsevich-Tschinkel's generalization in [KT19], rely heavily on the weak factorization theorem, and hence on resolution of singularities, it is unclear how to generalize Shinder's approach to positive characteristic. Similarly, Voisin's analysis [Voi19] requires resolution of singularities of the singular hypersurfaces with unramified cohomology constructed in [Sch19b], which in general is unknown in positive characteristic. Moreover, Voisin's approach did not allow to reprove Shinder's characteristic zero result in dimensions greater than nine, because it requires the knowledge of certain unramified cohomology groups of very general hypersurfaces, which seems out of reach in high dimensions.…”

“…Our usage of decompositions of the diagonal is more direct than in [Voi19] and we do in particular not use unramified cohomology. Instead, our approach relies on a moving lemma of Gabber, Liu and Lorenzini [GLL13, Theorem 2.3] and intersection theory on strictly semi-stable schemes using Fulton's work [Ful98].…”