Non-existence of negative curves

TL;DR

This paper constructs an infinite family of projective toric surfaces with Picard number one, blown up at a general point, whose Kleiman-Mori cones are not closed, providing insights into Nagata's conjecture and Seshadri constants.

Contribution

It presents new examples of toric surfaces with non-closed Kleiman-Mori cones, connecting to longstanding conjectures in algebraic geometry.

Findings

Existence of infinite families of such surfaces.

Kleiman-Mori cones are not closed in these examples.

Relation to Nagata's conjecture and Seshadri constants.

Abstract

Let $X$ be a projective toric surface of Picard number one blown up at a general point. We bring an infinite family of examples of such $X$ whose Kleiman-Mori cone of curves is not closed: there is no negative curve generating one of the two boundary rays of the cone. These examples are related to Nagata's conjecture and rationality of Seshadri constants.