The Stacks project

Lemma 55.5.7. Classification of proper subgraphs of the form

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

If $n > 5$, then given five $(-2)$-indices $h, i, j, k, l$ with $a_{hi}, a_{ij}, a_{jk}, a_{jl}$ nonzero, then up to ordering we have the $m$'s, $a$'s, $w$'s

  1. are given by

    \[ \left( \begin{matrix} m_1 \\ m_2 \\ m_3 \\ m_4 \\ m_5 \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & 0 & 0 & 0 \\ w & -2w & w & 0 & 0 \\ 0 & w & -2w & w & w \\ 0 & 0 & w & -2w & 0 \\ 0 & 0 & w & 0 & -2w \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ w \\ w \\ w \end{matrix} \right) \]

    with $2m_1 \geq m_2$, $2m_2 \geq m_1 + m_3$, $2m_3 \geq m_2 + m_4 + m_5$, $2m_4 \geq m_3$, and $2m_5 \geq m_3$.

Proof. See discussion above. $\square$

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