On Weak bounded negativity conjecture

TL;DR

This paper investigates bounds on the self-intersection numbers of integral curves on blown-up surfaces with effective anti-canonical divisors and proves weak bounded negativity for such curves in certain surface families.

Contribution

It provides new bounds on self-intersections on blown-up surfaces and establishes weak bounded negativity in a family of surfaces, advancing understanding of curve negativity.

Findings

Bounds on $C^2$ for integral curves on blow-ups of surfaces with effective $-K_X$

Proof of weak bounded negativity for integral curves in surface families

Extension of bounded negativity conjecture to specific surface classes

Abstract

In the first part of this article, we give bounds on self-intersections $C^2$ of integral curves $C$ on blow-ups $Bl_nX$ of surfaces $X$ with the anti-cannonical divisor $-K_X$ effective. In the last part, we prove the weak bounded negativity for self-intersections $C^2$ of integral curves $C$ in a family of surfaces $f:Y\longrightarrow B$ where $B$ is a smooth curve.