Corollary 11.18.5E2EF3C8673FC

Proof. Use the ruling section $S$ and fiber $f$ of lemma 11.13.1, and put $H=S+(3e+1)f$ and $P=4H-f$. Choose $E$ as in corollary 11.17.5, $\mu _ P$-stable and with $c_1(E)=4H$. The intersection numbers of lemma 11.13.1 give

By the same lemma, $P=4S+(12e+3)f$ is very ample. Take for $\omega $ the pulled-back Fubini–Study Kähler form representing $c_1(\mathcal O_ X(P))$. Degrees with respect to $\omega $ are intersection numbers with $P$, so $E$ is $\omega $-slope-stable with $\deg _\omega E>0$, and corollary 11.18.4 gives $3$-convexity, since $n=r=2$. By the choice of $E$, the complement is not $2$-convex. Finally, $1$-convexity would imply $2$-convexity, so the least index is three. $\square$