Pome510B6B95F8B4Lemma 10.28.2.510B6B95F8B4 The line bundle $A$ is generated by global sections. There is a semipositive metric $h_0$ on $L_0$ whose weight along $\ell $ is
where $w$ is a local equation of $\ell $. It is smooth off $\ell $ and has analytic singularities.
Proof. Consider the cubic forms
Their common zero set in $\mathbb P^2$ consists of the four central points. In affine coordinates at $p_1$, $p_2$ and $p_3$, their linear initial terms include $(z,y)$, $(z,x)$ and $(z,x-y)$, respectively, while at $p_4$ their quadratic initial terms include $x^2,xy,y^2$. These have no common zero on the corresponding exceptional projective lines. After the simple exceptional factors at $p_1,p_2,p_3$ and the double factor at $p_4$ are removed, the transformed cubics therefore generate $A$.
Their Fubini–Study metric $h_ A$ is smooth and semipositive. Let $h_\ell $ be the divisor metric on $\mathcal O(\ell )$. Since $2L_0=A+\ell $, the square root of $h_ A\otimes h_\ell $ defines a metric $h_0$ on $L_0$, with
The stated form of the weight and of its singularities follows directly. $\square$
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