Triple-point defective ruled surfaces (Q2477638)

Let \(S \subset \mathbb P^n\) be a smooth projective surface, \(K=K_S\) the canonical class and \(L\) the hyperplane divisor class on \(S\). The authors study a classical interpolation problem for the pair \((S,L)\), namely whether, for a general point \(p \in S\), the linear system \(|L-3p|\) has the expected dimension expdim\(|L-3p|\) \(=\) max\(\{-1,\) dim\(|L|-6\}\). If this is not the case, we call the pair \((S,L)\) triple-point defective. The paper under review continues from a previous paper by the same two authors: [``Triple-point defective regular surfaces'', 2007, \url{arXiv:0705.3912}], where some classification of triple-point defective pairs are classified. In both the papers the following assumptions hold: \(L-K\) very ample and \((L-K)^2 > 16\). The main result of the paper under review is the following Theorem. Suppose that the pair \((S,L)\) is triple-point defective, then \(S\) admits a ruling \(\pi:S \rightarrow C\), i.e. a morphism whose generic fibre is isomorphic to \(\mathbb P^1\). The same theorem was proved in the previous quoted previous paper under the hypothesis: \(S\) regular surface. In addition, a classification of line bundles \(L\) on a ruled surface \(S\), such that the pair \((S,L)\) is triple-point defective, is shown. The authors' method follows Reider's analysis of rank \(2\) bundles arising from points that do not impose independent conditions, adapted to the case of fat points, as explained by \textit{M. C. Beltrametti, P. Francia, A. J. Sommese} [Duke Math. J. 58, No. 2, 425--439 (1989; Zbl 0702.14010)].