Lemma 55.5.12 (0C8I)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 55: Semistable Reduction
Section 55.5: Classification of proper subgraphs
Lemma 55.5.12 (cite)

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Lemma 55.5.12. Nonexistence of proper subgraphs of the form

$\xymatrix{ \bullet \ar@{-}[r] & \bullet \ar@{-}[r] & \bullet \ar@{-}[r] \ar@{-}[d] & \bullet \ar@{-}[r] & \bullet \\ & & \bullet \ar@{-}[d] \\ & & \bullet }$

Assume $n > 7$ . There do not exist $7$ distinct $(-2)$ -indices $f, g, h, i, j, k, l$ such that $a_{fg}, a_{gh}, a_{ij}, a_{jh}, a_{kl}, a_{lh}$ are nonzero.

Proof. See discussion above. $\square$

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