Proposition 55.7.4. Let $g \geq 2$. For every numerical type $T$ of genus $g$ and prime number $\ell > 768g$ we have
where $\mathop{\mathrm{Pic}}\nolimits (T)$ is as in Definition 55.4.1. If $T$ is minimal, then we even have
where $g_{top}$ as in Definition 55.3.11.
Proof. Say $T$ is given by $n, m_ i, a_{ij}, w_ i, g_ i$. If $T$ is not minimal, then there exists a $(-1)$-index. After replacing $T$ by an equivalent type we may assume $n$ is a $(-1)$-index. Applying Lemma 55.4.4 we find $\mathop{\mathrm{Pic}}\nolimits (T) \subset \mathop{\mathrm{Pic}}\nolimits (T')$ where $T'$ is a numerical type of genus $g$ (Lemma 55.3.9) with $n - 1$ indices. Thus we conclude by induction on $n$ provided we prove the lemma for minimal numerical types.
Assume that $T$ is a minimal numerical type of genus $\geq 2$. Observe that $g_{top} \leq g$ by Lemma 55.3.14. If $A = (a_{ij})$ then since $\mathop{\mathrm{Pic}}\nolimits (T) \subset \mathop{\mathrm{Coker}}(A)$ by Lemma 55.4.3. Thus it suffices to prove the lemma for $\mathop{\mathrm{Coker}}(A)$. By Lemma 55.7.3 we see that $m_ i|a_{ij}| \leq 768g$ for all $i, j$. Hence the result by Lemma 55.2.6. $\square$
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