Existence of curves with prescribed topological singularities (Q2781387)

The question addressed in the paper is: Given a linear system on a given complex algebraic surface, does there exist an irreducible algebraic curve in this linear system which has a collection of singular points prescribed up to topological equivalence? An answer is obtained in the form of a sufficient existence criterion for such a curve. The criterion involves only numerical invariants of singularities, linear system and a surface, and is applicable to arbitrary singularities and ample linear systems on various classes of surfaces. Furthermore, this sufficient criterion reveals asymptotics comparable with the asymptotics of necessary conditions, when the self-intersection of the curves considered grows to infinity. The present paper heavily relies on the paper by \textit{G.-M. Greuel, Ch. Lossen} and \textit{E. Shustin} [Invent. Math. 133, No. 3, 539-580 (1998; Zbl 0924.14013)] where the same question for plane curves have been answered in a similar way. However, the authors use a different technical tool, the Kodaira vanishing theorem as was suggested by \textit{G. Xu} [J. Reine Angew. Math. 469, 199-209 (1995; Zbl 0833.14028)].