Generators for the defining ideal of certain rational surfaces (Q807703)

One considers the rational surfaces \({\mathbb{P}}^ 2(Z)\) obtained by blowing-up the projective plane in the points of a finite set of points, Z. If \(E_ 1,...,E_ s\) are the (divisor classes on \({\mathbb{P}}^ 2(Z)\) of the) exceptional lines and \(E_ 0\) (the divisor class of) the proper transform of a line missing all the points of Z, then there is an integer \(\sigma\) which depends only on the Hilbert function of the ideal of Z in \({\mathbb{P}}^ 2\), such that \(tE_ 0-\sum E_ i\quad (t\geq \sigma)\) is very ample iff no \(\sigma\) elements of Z lie on a line of \({\mathbb{P}}^ 2\) [cf. \textit{E. D. Davis} and \textit{A. V. Geramita}, Math. Ann. 279, No.3, 435-448 (1988; Zbl 0657.14003)]. The main result of this paper says that for \(t\geq \sigma +1\) the ideals of the images by these embeddings are generated by quadrics.