Relative canonical sheaves of a family of curves (Q1775535)

Let \(f: X\to S\) be a relatively minimal nonsingular surface over a nonsingular curve over \(\mathbb{C}\) whose general fiber has genus \(g\geq 2\). The author considers the relative canonical algebra \({\mathcal R}(f)= \bigoplus_{m\geq 0} f_*(\omega^{\otimes m}_{X/S})\). Miles Reid conjectured that \({\mathcal R}\) is generated in degree \(\leq 3\) as an \({\mathcal O}_S\)-algebra (1-2-3 Conjecture). K. Konno and M. Mendes Lopes presented some counterexamples which need a generator in degree 4. \textit{K. Konno} [J. Reine Angew. Math. 533, 171--205 (2001; Zbl 0965.14004)] showed that \({\mathcal R}(f)\) is generated in degree \(\leq 4\) and pointed out when it needs a generator in degree 4. In the paper under review, the author studies \({\mathcal R}(f)\) when \(X\), \(S\) and \(f\) are not necessarily over \(\mathbb{C}\). The main result is to show that \({\mathcal R}(f)\) is generated in degree \(\leq 5\). The author uses Koszul cohomology along with the global generatedness to conclude that \({\mathcal R}(f)\) is generated in degree \(\leq 5\).