The Stacks project

Lemma 55.5.4. Classification of proper subgraphs of the form

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

If $n > 4$, then given four $(-2)$-indices $i, j, k, l$ with $a_{ij}, a_{ik}, a_{il}$ nonzero, then up to ordering we have the $m$'s, $a$'s, $w$'s

  1. are given by

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

    with $2m_1 \geq m_2 + m_3 + m_4$, $2m_2 \geq m_1$, $2m_3 \geq m_1$, $2m_4 \geq m_1$. Observe that this implies $m_1 \geq \max (m_2, m_3, m_4)$.

Proof. See discussion above. $\square$

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