Weak Bounded Negativity Conjecture

TL;DR

This paper proves a version of the Weak Bounded Negativity Conjecture, establishing that on complex smooth projective surfaces, the self-intersection number of certain reduced curves is bounded below when their components' genera are bounded.

Contribution

The paper provides a proof of the Weak Bounded Negativity Conjecture for complex smooth projective surfaces, linking genus bounds to intersection number bounds.

Findings

Self-intersection numbers are bounded below under genus constraints

The result applies to reduced curves with components of bounded genus

Advances understanding of negativity properties in algebraic geometry

Abstract

In this paper, we prove the following "Weak Bounded Negativity Conjecture", which says that given a complex smooth projective surface $X$, for any reduced curve $C$ in $X$ and integer $g$, assume that the geometric genus of each component of $C$ is bounded from above by $g$, then the self-intersection number $C^2$ is bounded from below.