Imposing singular points and many nodes to curves on surfaces
Given a very ample divisor \(H\) on a smooth projective algebraic surface \(S\) over a field of characteristic \(0\), the authors give an explicit lower bound to \(d\) such that a collection of isolated singularities with prescribed topological types, located in \(S\) in general position, imposes independent conditions to the linear system \(|dH|\). More precisely, with an isolated singular curve point one can associate a zero-dimensional scheme (for instance, the scheme associated with an ordinary \(m\)-fold point is defined by the \(m\)-th power of the maximal ideal in the local ring of the surface), and the main theorem claims that \(H^1(S,{\mathcal J}(d))=0\) for the ideal sheaf \({\mathcal J}\) of a ``generic'' zero-dimensional scheme associated with the given multi-singularity. The bound to \(d\) becomes sharper when most of singularities are ordinary nodes. The proof is based on the so-called ``Horace method'', invented by \textit{A. Hirschowitz} [Manuscripta Math. 50, 337-388 (1985; Zbl 0571.14002)]. Under some extra conditions, there exists an irreducible curve in \(|dH|\) having exactly the prescribed collection of singularities. Reviewer's remark. Another approach to the problem of existence of curves with prescribed singularities in given linear systems on smooth algebraic surfaces, based on the Kodaira vanishing theorem, was suggested by \textit{T. Keilen} and \textit{I. Tyomkin} [Trans. Am. Math. Soc. 354, No. 5, 1837-1860 (2002; Zbl 0996.14013)].
scientific article; zbMATH DE number 1753106