A Property of Upper Level Sets of Lelong Numbers of Currents on ${\mathbb{P}}^2$

Let $T$ be a positive closed current of bidimension $(1,1)$ with unit mass on the complex projective space ${\mathbb{P}}^2$. For $\alpha >2\mathrm{/}5$ and $\beta = (2-2\alpha)/3$weshowthatif$T$ has four point with Lelong number greater than $\alpha$, the upper level set $E_{\beta }^{+}(T)$ of points of $T$ with Lelong number strictly larger than $\beta$ is contained within a conic with the exception of at most one point.