Birational morphisms to \({\mathbb{P}}^ 2:\) An ideal-theoretic perspective (Q1110591)

One of the result of the paper is the following theorem: Let Z be a finite set of points in \({\mathbb{P}}^ 2(k)\) (k is an algebraically closed field of an arbitrary characteristic) and \(\pi:\quad Q\to {\mathbb{P}}^ 2(k)\) the blow-up of Z, and let \(E=\pi^{-1}(Z)\) be the exceptional divisor of \(\pi\). Let L be a general line in \({\mathbb{P}}^ 2(k)\) and \({\mathcal I}_ Z\) the ideal of Z in \({\mathcal O}_{{\mathbb{P}}^ 2(k)}\). Suppose that \(h^ 1({\mathcal I}_ Z(n-1))=0\). Then \(\pi^*(nL)-E\) is a very ample divisor if and only if no line contains n points of Z. The whole paper is devoted to generalizations in various aspects of this result to the case when \(\pi\) is a birational morphism from Q to P, both smooth complete algebraic surfaces. Many applications of these results in obtaining examples of algebraic surfaces are also given.