On the Bounded Negativity Conjecture and singular plane curves

TL;DR

This paper discusses the Bounded Negativity Conjecture in algebraic geometry, proposing new characteristic-free conjectures and establishing bounds on the negativity of rational curves with limited singularities.

Contribution

It introduces characteristic-free conjectures related to negativity bounds and provides explicit bounds for the H-constant of rational curves with few singular points.

Findings

H-constant of rational curves with ≤9 singular points exceeds -2

Proposes characteristic-free versions of the Negativity Conjecture

Develops bounds on numerical characteristics of curves

Abstract

There are no known failures of Bounded Negativity in characteristic 0. In the light of recent work showing the Bounded Negativity Conjecture fails in positive characteristics for rational surfaces, we propose new characteristic free conjectures as a replacement. We also develop bounds on numerical characteristics of curves constraining their negativity. For example, we show that the $H$-constant of a rational curve $C$ with at most $9$ singular points satisfies $H(C)>-2$ regardless of the characteristic.