Lemma 10.28.7A52A7B9951F1

Proof. First blow up $p_4$ in $\mathbb P^2$. The map $[x:y]$ then becomes holomorphic: in the chart $x=r$, $y=rs$ with $z=1$, it is $[1:s]$, and the other chart covers the remaining direction. Its fibers are the strict transforms of the lines through $p_4$, and the pullback of $\mathcal O_{\mathbb P^1}(1)$ is $H-E_4$. Next blow up the other three central points, which lie in the directions $a=\infty ,0,1$, respectively. Each affected fiber becomes the union of its strict transform and one exceptional line meeting it, while every other fiber remains a projective line. This proves connectedness of the fibers and the formula for $P$. The equality $L_0=\mathcal O(\ell )+P$ follows from the divisor classes.

In this identification, the restrictions of $F_1,F_2$ to $X_0$ are $s_\ell \otimes f^*x$ and $s_\ell \otimes f^*y$, where $x$ and $y$ are regarded as sections of $\mathcal O_{\mathbb P^1}(1)$. Indeed, at $t=0$ the forms are $xz$ and $yz$; the pullback of $\{ z=0\} $ is $\ell +E_1+E_2+E_3$, and removing the four exceptional factors from the transforms of $xz$ and $yz$ leaves exactly these sections. In a local frame of $L$, the weight $\log (|F_1|^2+|F_2|^2)$ defines a semipositive metric, since a change of frame multiplies both coefficients by the same nonvanishing holomorphic function. On every fiber $X_ t$ with $t\ne 0$ the two sections have no common zero, so the weight is not identically $-\infty $ there. On the central fiber the weight is the sum of the divisor weight of $\ell $ and the pullback of the Fubini–Study weight, which proves the claim and the curvature formula. $\square$