A combinatorial proof of non-speciality of systems with at most 9 imposed base points

Abstract

It is known that the Segre-Gimigliano-Harbourne-Hirschowitz Conjecture holds for linear systems of curves with at most 9 imposed base fat points. We give a nice proof based on a combinatorial method of showing non-speciality of such systems. We will also prove, by the same method, that systems $ℒ(km;m^{xk^2})$ and $ℒ(km +1; m^{xk^2})$ are non-special.