Conditions imposed by tacnodes and cusps

Abstract

The study of linear systems of algebraic plane curves with fixed imposed singularities is a classical subject which has recently experienced important progress. The Horace method introduced by A. Hirschowitz has been successfully exploited to prove many H1-vanishing theorems, even in higher dimension. Other specialization techniques, which include degenerations of the plane, are due to Z. Ran and C. Ciliberto and R. Miranda. G. M. Greuel, C. Lossen and E. Shustin use a local specialization procedure together with the Horace method to give the first asymptotically proper general existence criterion for singular curves of low degree. In this paper we develop a specialization method which allows us to compute the dimension of several linear systems as well as to substantially improve the bounds given by Greuel, Lossen and Shustin for curves with tacnodes and cusps.