Upper level sets of Lelong numbers on Hirzebruch surfaces

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 $\beta\geq (\alpha + (a+1)\alpha^{\prime})/(a+2)$.