The Stacks project

Proposition 55.5.17. Let $n, m_ i, a_{ij}, w_ i, g_ i$ be a numerical type of genus $g$. Let $I \subset \{ 1, \ldots , n\} $ be a proper subset of cardinality $\geq 2$ consisting of $(-2)$-indices such that there does not exist a nonempty proper subset $I' \subset I$ with $a_{i'i} = 0$ for $i' \in I$, $i \in I \setminus I'$. Then up to reordering the $m_ i$'s, $a_{ij}$'s, $w_ i$'s for $i, j \in I$ are as listed in Lemmas 55.5.1, 55.5.2, 55.5.3, 55.5.4, 55.5.5, 55.5.7, 55.5.8, 55.5.9, 55.5.10, 55.5.13, or 55.5.14.

Proof. This follows from the discussion above; see discussion at the start of Section 55.5. $\square$

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