Partially negative cycles and projective embeddings of surfaces of general type (Q1976571)

Let \(S\) be a nonsingular minimal complex surface of general type. By a famous result of \textit{E. Bombieri} [Publ. Math., Inst. Hautes Étud. Sci. 42, 171-219 (1973; Zbl 0259.14005)], the linear system \(|5K_S|\) induces a birational morphism that is an isomorphism except that it contracts all the rational curves with self-intersection \(-2\) (so it induces an embedding of the canonical model of \(S\)). On the other hand, \textit{J. Yang} [in: Algebraic geometry and algebraic number theory, Pap. Spec. Year algebraic geometry Tianjun 1989/1990, Nankai Ser. Pure Appl. Math. Theor. Phys. 3, 173-178 (1992; Zbl 0901.14021)] showed that for a given divisor \(W\) (the minimally negative cycle), for sufficiently large \(m\), the linear system \(|mK_S-W|\) gives a projective embedding of \(S\). In the paper under review, the author shows that one can construct intermediated embeddings between the two above. Concretely, take a non-singular minimal complex surface of general type \(S\), and choose a subset of the \((-2)\)-curves of \(S\). Then the author constructs a cycle \(X\) that has intersection \(0\) with each of the chosen curves, and negative with all the other \((-2)-\)curves. Then he shows that, for \(m\) large enough, \(|mK_S-X|\) defines a birational morphism that contracts the chosen curves, and is an embedding everywhere else: Main tool here is \textit{I. Reider}'s theorem [Ann. Math. (2) 127, 309-316 (1988; Zbl 0663.14010)].