On the relative minimal model program for threefolds in low characteristics (Q2089077)

The Minimal Model Program has been settled for threefolds defined over fields of positive characteristic \(p>5\). The proof consists of two main parts: the existence of divisorial contractions, which relies on \textit{S. Keel}'s work in [Ann. Math. (2) 149, No. 1, 253--286 (1999; Zbl 0954.14004)] and holds in every characteristic, and the existence of flips, proven by \textit{C. D. Hacon} and \textit{C. Xu} in [J. Am. Math. Soc. 28, No. 3, 711--744 (2015; Zbl 1326.14032)], using techniques specific to characteristic \(p>5\) coming from the existence of certain Frobenius splitting. In this article, the authors establish the existence of flips necessary to run a birational MMP for 3-folds in every characteristic. The main technical result for the existence of flips is Proposition 3.3, where techniques from the theory of \(F\)-singularities are used to show the existence of 1-complemented flips. As a consequence of the MMP, the existence of dlt (divisorial log terminal) modifications and Kollár's components for \(\mathbb{Q}\)-factorial 3-folds are established, thus implying inversion of adjunction and \((\mathcal{O}_X)\)-rationality of klt (Kawamata log terminal) 3-fold singularities in every characteristic. Finally, the semi-stable MMP for families of surfaces in every characteristic is also established.