Conditions imposed by tacnodes and cusps

The study of linear systems of algebraic plane curves with fixed imposed si ngularities is a classical subject which has recently experienced important progress. The Horace method introduced by A. Hirschowitz has been successf ully exploited to prove many H-1-vanishing theorems, even in higher dimensi on. Other specialization techniques, which include degenerations of the pla ne, are due to Z. Ran and C. Ciliberto and R. Miranda. G. M. Greuel, C. Los sen and E. Shustin use a local specialization procedure together with the H orace method to give the first asymptotically proper general existence crit erion for singular curves of low degree. In this paper we develop a special ization method which allows us to compute the dimension of several linear s ystems as well as to substantially improve the bounds given by Greuel, Loss en and Shustin for curves with tacnodes and cusps.