Very ample line bundles on regular surfaces obtained by projection
Let \(X \subset \mathbb P^N\) be a smooth projective algebraic surface defined over an algebraically closed field, with hyperplane bundle \(\mathcal L = \mathcal O_X(1)\), degree \(d\), and sectional genus \(g\). Let \(\sigma: \widehat{X} \to X\) be the blowing-up at distinct points \(x_1, \dots , x_m \in X\) and consider the line bundle \(\widehat{\mathcal L}:= \sigma^* \mathcal L \otimes \mathcal O_{\widehat{X}} (-E_1 - \dots -E_m)\), where \(E_i\) is the exceptional divisor corresponding to \(x_i\). The problem of finding conditions for the very ampleness of \(\widehat{\mathcal L}\) in terms of the configuration of points \(x_1, \dots , x_m\) has been considered by many authors, e. g., \textit{E. M. Bese} [Math. Ann. 262, 225--238 (1983; Zbl 0491.14009)], \textit{F. Catanese} and \textit{M. Franciosi} [Proc. Hirzebruch 65 Conf. Algebraic Geometry, Ramat Gan, 1993, Isr. Math. Conf. Proc. 9, 109--140 (1996; Zbl 0855.14004)], \textit{M. Coppens} [Pac. J. Math. 202, No. 2, 313--327 (2002; Zbl 1059.14011)]. In the paper under review the author proves the following. Suppose that \(h^1(X, \mathcal O_X)=0\) and \(m \leq d-2g-1\). Then \(\widehat{\mathcal L}\) is very ample if and only if for all \(l\), \(1 \leq l \leq m\), any \(l\) distinct points of \(\{x_1, \dots ,x_m\}\) do not lie on any rational normal curve of degree \(l\) contained in \(X\). A key role in the proof is played by inner projections of \(X\) from a subset of \(\{x_1, \dots ,x_m\}\). As an application, the author determines precise conditions ensuring the very ampleness of some line bundles arising in the classification of surfaces of sectional genus \(2\) and \(3\), namely rational conic fibrations (\(g=2\) and \(3\)) and Bordiga surfaces (\(g=3\)).
scientific article; zbMATH DE number 5115364
scientific article