On non-rational numerical Del Pezzo surfaces
In this paper the author studies the numerical Del Pezzo surfaces. By definition a numerical Del Pezzo surface is a compact complex Moishezon surface \(X\) whose anti-canonical (Weil) divisor \(-K_X\) satisfies the following two conditions: (i) \((-K_X)^2>0\), and (ii) \((-K_X.C)>0\) for every irreducible curve \(C\) on \(X\). -- This class of surfaces generalizes the class of classical Del Pezzo surfaces. By using an algebraicity criterion of L. Brenton, one sees imediately that every numerical Del Pezzo surface is projective algebraic. These surfaces have been considered in the early eighties by \textit{F. Sakai} [Math. Ann. 269, 389-410 (1984; Zbl 0533.14016)] and by \textit{L. Bădescu} [Ann. Univ. Ferrara, Nuova Ser., Sez. VII 29, 165-177 (1983; Zbl 0544.14023)], at least in the case when they have only \(\mathbb{Q}\)-Gorenstein singularities. The author first proves a slight improvement of a result of \textit{L. Bădescu} [which however also follows from the results of another paper by \textit{L. Bădescu}, J. Reine Angew. Math. 367, 76-89 (1986; Zbl 0604.14020)]. Then he defines a notion of minimal contraction of a ruled surface and proves a criterion for the minimal contraction of some ruled surfaces to be a numerical Del Pezzo surface. Finally he defines the notion of index of non-rational numerical Del Pezzo surfaces and studies its properties.
index of non-rational numerical Del Pezzo surfaces
scientific article; zbMATH DE number 871824