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

TL;DR

This paper explores how the Segre Problem influences the structure of the Mori cone in blown-up surfaces, linking linear systems and cone geometry, and proposing a generalized conjecture.

Contribution

It introduces a generalized Segre Problem for surfaces and proves a conditional relationship between the Segre Problem and the Mori cone structure.

Findings

If the Segre Problem holds, part of the Mori cone matches the positive cone.

Generalization of SHGH Conjectures to arbitrary surfaces.

Extension of de Fernex's result relating linear systems and cone structures.

Abstract

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 $X$.