Embeddings of blown-up plane
Let \(X\) be a smooth projective surface defined over the complex number field \(\mathbb{C}\). Then there is the following problem which is very interesting: Problem: If \(X\) is a blown up surface of a smooth surface \(S\), then study the property of the pullback bundle \(L\) of a line bundle \(A\) on \(S\). \textit{G. Xu} [Manuscr. Math. 86, 195-197 (1995; Zbl 0836.14004) and J. Reine Angew. Math. 469, 199-209 (1995; Zbl 0833.14028)], \textit{O. Küchle} [Math. Ann. 304, No. 1, 151-155 (1996; Zbl 0834.14024)], and \textit{K. Yokoyama} [J. Math. Soc. Japan 51, No. 3, 523-533 (1999)] have given some criteria for the pullback bundle of ample line bundle to be ample. \textit{J. D'Almeida} and \textit{A. Hirschowitz} [Math. Z. 211, No. 3, 479-483 (1992; Zbl 0759.14004)] considered the case of points in generic position on \(\mathbb{P}^{2}\), and gave a criterion for the pullback bundle to be very ample. In this paper, the author considers the above problem in the case in which \(S\) is a projective space \(\mathbb{P}^{2}\) and gives a criterion for the pullback to be \(k\)-very ample. Here the notion of \(k\)-very ampleness of a line bundle \(L\) is defined by the following: For any \(0\)-dimensional subscheme \(Z\) of length \(k+1\) on \(S\), the map \(H^{0}(L)\to H^{0}(L_{Z})\) is surjective. By definition \(0\)-very-ampleness (resp. \(1\)-very ampleness) is equivalent to spannedness (resp. very ampleness). So the criterion in this paper is a generalization of the paper by D'Almeida and Hirschowitz cited above. Though the criterion in this paper is true, there are many mistakes and you need to read this paper carefully.
scientific article; zbMATH DE number 1335049