The Stacks project

Lemma 55.4.5. Let $n, m_ i, a_{ij}, w_ i, g_ i$ be a numerical type $T$. If the genus $g$ of $T$ is $\leq 0$, then $\mathop{\mathrm{Pic}}\nolimits (T) = \mathbf{Z}$.

Proof. By induction on $n$. If $n = 1$, then the assertion is clear. If $n > 1$, then $T$ is not minimal by Lemma 55.3.13. After replacing $T$ by an equivalent type we may assume $n$ is a $(-1)$-index. By Lemma 55.4.4 we find $\mathop{\mathrm{Pic}}\nolimits (T) \subset \mathop{\mathrm{Pic}}\nolimits (T')$. By Lemma 55.3.9 we see that the genus of $T'$ is equal to the genus of $T$ and we conclude by induction. $\square$

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