Unbounded negativity on rational surfaces in positive characteristic

TL;DR

This paper demonstrates that the Bounded Negativity Conjecture does not hold for rational surfaces in positive characteristic by constructing explicit examples with arbitrarily negative self-intersection curves.

Contribution

It provides explicit constructions of rational surfaces in positive characteristic with unbounded negative self-intersection, disproving the conjecture in this setting.

Findings

Explicit blowups contain smooth rational curves with arbitrarily negative self-intersection.

The Bounded Negativity Conjecture fails for rational surfaces in positive characteristic.

Constructs counterexamples to a long-standing conjecture in algebraic geometry.

Abstract

We give explicit blowups of the projective plane in positive characteristic that contain smooth rational curves of arbitrarily negative self-intersection, showing that the Bounded Negativity Conjecture fails even for rational surfaces in positive characteristic.