On polarized surfaces of degree three whose adjoint bundles are not spanned (Q1898872)

Let \((S,L)\) be a pair where \(S\) is a smooth complex surface and \(L\) is a line bundle on \(S\). If \(L\) is very ample it is a classical result of Sommese and Van de Ven that \(K + L\) is generated by global sections as soon as it has any. If \(L\) is only ample and spanned, Reider's theorem implies that \(K + L\) is spanned unless \((S,L)\) is a scroll under the assumption \(L^2 \geq 5\). This work investigates what happens below Reider. In particular pairs \((S,L)\) with \(L\) ample and spanned, \(L^2 = 3\) and \(K + L\) not spanned are investigated. These surfaces are expressed by \(|L |\) as a triple cover of \(\mathbb{P}^2\). Several general results on these surfaces are obtained. In the case in which \(\text{Kod} (S) \leq 0\) a detailed analysis of the possible numerical invariants of these pairs is conducted.