Upper level sets of Lelong numbers on Hirzebruch surfaces

Abstract

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