Surfaces of sectional genus \(\leq 8\) with no trisecant lines
Let \(S\) be a smooth projective surface over \(\mathbb{C}\) and \(L\) a line bundle on \(S\). According to a paper by \textit{M. Beltrametti} and \textit{A. J. Sommese} [``Zero cycles and \(k\)-th order embeddings of smooth projective surfaces'', Proc. Projective Surfaces and their Classification, Cortona (1988), Symp. Math., INDAM] we say that \(L\) is \(k\)-very ample, for \(k \geq 2\), if for any 0-dimensional subscheme \((Z,{\mathcal O}_ Z)\) of length \(k+1\) the restriction map \(\Gamma (L) \to \Gamma (L \bigotimes {\mathcal O}_ Z)\) is a surjection. For \(k=1\) this means that \(L\) is very ample; for \(k=2\) on the other hand it means that if \(S \subset \mathbb{P}^ N\) is the embedding defined by the global sections of \(L\) (i.e. \(L={\mathcal O} (1)_{| S})\) then there are no 0-dimensional subscheme of length 3 on a line; in this case we will say that \(S \subseteq \mathbb{P}^ N\) has no 3-secant lines. A classical problem, if \(L\) is very ample, is to give a classification of the pairs \((S, L) \) with low numerical invariants, for instance with low sectional genus. -- In this paper we want to classify the pair \((S,L)\) with sectional genus \(\leq 8\) and such that \(L\) is \(k\)-very ample for \(k \geq 2\). For \(g \leq 5\) the classification was carried out in the paper by \textit{M. Beltrametti} and \textit{A. J. Sommese} in Algebraic Geometry, Proc. Int. Conf., L'Asquila 1988, Lect. Notes Math. 1417, 24-51 (1990; Zbl 0706.14007) and for \(g=6\) by \textit{A. Lanteri} [``2-spanned surfaces of sectional genus six'', Ann. Mat. Pura Appl., IV. Ser. 165, 197-216 (1993)]. We therefore consider the cases \(g=7\) and 8.
classification of \(k\)-very ample line bundles
scientific article; zbMATH DE number 571900
scientific article