PomeProposition 11.13.4.B6B9C2094631 For $F_ s=L_ s^*$, $s=1,2$, there are Kähler forms $\eta _ s$ with $\deg _{\eta _ s}F_ s=1$. Consequently, the split bundle $E_0^*=F_1\oplus F_2$ satisfies the hypotheses of theorem 11.9.4.
Proposition 11.13.4B6B9C2094631
Proof. The integral classes
\[ A_1=6S+(6e+1)f,\qquad A_2=S+(13e+6)f \]
are very ample by lemma 11.13.1. Pullbacks of normalized Fubini–Study forms under the corresponding embeddings give Kähler forms $\eta _ s$ representing $c_1(\mathcal O_ X(A_ s))$. A direct computation of intersection numbers gives
\[ c_1(L_1)\cdot A_1=-1,\qquad c_1(L_2)\cdot A_2=-1, \]
so the degrees of the duals are equal to one. Since $n=2$, the closed positive $(1,1)$-forms $\eta _ s$ are admissible test forms $\Theta _ s$ in theorem 11.9.4. $\square$
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