Counterexamples to Fujita's conjecture on surfaces in positive characteristic (Q2125995)

\textit{T. Fujita} proposed the following conjecture in [Adv. Stud. Pure Math. 10, 167--178 (1987; Zbl 0659.14002)]: Fujita's freeness conjecture. Let \(X\) be a smooth projective variety of dimension \(n\) over an algebraically closed field of characteristic zero and \(A\) an ample divisor on \(X\). Then \(|K_X+mA|\) is free if \(m\geqq n+1\). In the paper under review, the authors study the conjecture for surfaces in positive characteristic and give counterexamples to it. The main result is as follows. Theorem 1.2. Let \(\mathbf{k}\) be an arbitrary algebraically closed field of positive characteristic and and \(m \in \mathbb{N}_+\) an arbitrary positive integer. Then there exists a smooth projective surface \(S\) over \(\mathbf{k}\) admitting an ample divisor \(A\) such that \(|K_S+mA|\) is not free. In the above theorem, the surface \(S\) is given by a generalization of \textit{M. Raynaud}'s construction in [Tata Inst. fundam. Res., Stud. Math. 8, 273--278 (1978; Zbl 0441.14006)].