Uniform bounds for strongly \(F\)-regular surfaces (Q2790704)

The aim of this paper is to study the relationship between Kawamata log terminal (klt) pairs and strongly \(F\)-regular pairs for positive characteristic surface. Given a set \(I\subseteq[0,1]\), there is an associated hyperstandard set \(D(I)\) (see Definition 2.1), which arises naturally as the coefficients of divisors in the adjunction formula (Lemma 2.3). The author proved that if \(I\subseteq(0,1)\cap\mathbb{Q}\) is a finite subset, then there exists a positive constant \(p_0\) depending only on \(I\) such that if \((X, B)\) is a two dimensional klt pair defined over an algebraically closed field of characteristic \(p>p_0\) and such that the coeffeicients of \(B\) belongs to \(D(I)\), then \((X, B)\) is strongly \(F\)-regular. It is worth to point out that when \(B=0\), this is a result of \textit{N. Hara} [Adv. Math. 133, No. 1, 33--53, Art. No. AI971682 (1998; Zbl 0905.13002)] and in this case \(p_0=5\). Using this result and recent work of \textit{C. D. Hacon} and \textit{C. Xu} [J. Am. Math. Soc. 28, No. 3, 711--744 (2015; Zbl 1326.14032)], the author showed the existence of dlt flips: there exists \(p_I\) depending only on \(I\) such that if \((X,B)\) is a three dimensional dlt pair over an algebraically closed field of characteristic \(p>p_I\) and the coefficients of \(B\) belongs to \(I\), and \(f\): \(X\to Y\) is a \((X, B)\)-flipping contraction, then flip exist. Recently \textit{C. Birkar} [Ann. Sci. Éc. Norm. Supér. (4) 49, No. 1, 169--212 (2016; Zbl 1346.14040)] has obtained a stronger version of this result by using different methods.