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

Abstract

We propose a generalization of SHGH Conjectures to a smooth projective surface Y: the so-called Generalized Segre Conjecture. The study of linear systems on Y can be translated in terms of the Mori cone of the blow-up $$X=\mathrm{Bl }_r Y$$ at $$r$$ general points. A consequence of the Generalized Segre Conjecture is the so-called List Conjecture, a generalization of the Bounded Negativity Conjecture. Generalizing a result from [7], we prove that if the List Conjecture holds true, then a part of $$\overline{\mathrm{NE }}(X)$$ does coincide with a part of the positive cone of $$X$$ .