Upper level sets of Lelong numbers on Hirzebruch surfaces

TL;DR

This paper investigates the structure of upper level sets of Lelong numbers for positive closed (1,1)-currents on Hirzebruch surfaces, revealing their containment within specific algebraic curves depending on the surface and current parameters.

Contribution

It provides new bounds and geometric descriptions of Lelong number level sets on Hirzebruch surfaces, extending previous results known for projective planes.

Findings

For f1=0, the level set is contained in a degree 2 curve, possibly missing one point.

For general f1, the level set is contained in either a (0,1) curve or (a+1) (1,0) curves.

The paper establishes bounds on f1 for the containment of level sets in specific algebraic curves.

Abstract

Let $\mathbb F_a$ denote the Hirzebruch surfaces and $\mathcal{T}_{\alpha,\alpha^{\prime}}(\mathbb{F}_{a})$ denotes the set of positive, closed $(1,1)$-currents on $\mathbb{F}_{a}$ whose cohomology class is $\alpha F+\alpha^{\prime} H$ where $F$ and $H$ generates the Picard group of $\mathbb F_a$. $E^+_{\beta}(T)$ denotes the upper level sets of Lelong numbers $\nu(T,x)$ of $T\in \mathcal{T}_{\alpha,\alpha^{\prime}}(\mathbb{F}_{a})$. When $a=0$, ($\mathbb F_a=\mathbb P^1\times \mathbb P^1$), for any current $T\in \mathcal T_{\alpha,\alpha'}(\mathbb P^1\times \mathbb P^1)$, we show that $E^{+}_{(\alpha+\alpha')/3}(T)$ is contained in a curve of total degree $2$, possibly except $1$ point. For any current $T\in \mathcal T_{\alpha,\alpha'}(\mathbb F_a)$, we show that $ E^{+}_{\beta}(T)$ is contained in either in a curve of bidegree $(0,1)$ or in $a+1$ curves of bidegree $(1,0)$ where $\beta\geq (\alpha + (a+1)\alpha^{\prime})/(a+2)$.