On a question of Miles Reid
Let \(f:S\to B\) be a relatively minimal fibration, where \(S\) (respectively \(B)\) is a non-singular projective surface (respectively curve) and suppose that the general fibre of \(f\) is a smooth, irreducible curve of genus \(g\geq 2\). Let \({\mathcal R}(f)= \bigoplus_{n\geq 0} f_*(K^{\otimes n}_{S |B})\) the relative canonical algebra. Some years ago \textit{M. Reid} [``Problems on pencils of small genus'' (Preprint 1990)] asked whether \({\mathcal R}(f)\) is always generated in degree 1, 2 and 3. In this paper the authors present explicit counter-examples (of hyperelliptic and non-hyperelliptic fibrations of unbounded genus) to the above conjecture.
scientific article; zbMATH DE number 1355507