On the influence of the Segre Problem on the Mori cone of blown-up surfaces

Fulvio Di Sciullo

Published 2011-10-04, updated 2012-06-18Version 3

We propose a generalization of SHGH Conjectures to a smooth projective surface Y: the so called Segre Problem. The study of linear systems on Y can be translated in terms of the Mori cone of the blow up $X = Bl_r Y$ at $r$ general points. Generalizing a result by de Fernex, we prove that if Segre Problem holds true, then a part of the Mori cone of $X$ does coincide with a part of the positive cone of $X$.