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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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