On complex projective surfaces with trigonal hyperplane sections
Let \((S,L)\) be a polarized surface, i.e., \(S\) is a complex smooth projective surface and \(L\) is an ample line bundle on it. Assume that \(L\) is generated everywhere by its global sections. A main result of the present article is that, with the additional hypothesis \((L^ 2)\geq 10\), we have \((K_ S+L\cdot K_ S)\leq -3\) and \(| L|\) contains a trigonal curve \(C\) of genus \(\geq 4\) if and only if either \((S,L)\) is a rational surface ruled by cubics or the \(g^ 1_ 3\) of \(C\) is cut out by \(| -K_ S|\). If \(L\) is very ample and \(S\) is regular this is a result of \textit{F. Serrano} [``Surfaces having a hyperplane section with a special pencil'' (preprint)]; the hypothesis \((L^ 2)\geq 10\) is not necessary in Serrano's result.
scientific article; zbMATH DE number 4160932