The Stacks project

Lemma 55.5.1. Classification of proper subgraphs of the form

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

If $n > 2$, then given a pair $i, j$ of $(-2)$-indices with $a_{ij} > 0$, then up to ordering we have the $m$'s, $a$'s, $w$'s

  1. are given by

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

    with $w$ arbitrary and $2m_1 \geq m_2$ and $2m_2 \geq m_1$, or

  2. are given by

    \[ \left( \begin{matrix} m_1 \\ m_2 \end{matrix} \right), \quad \left( \begin{matrix} -2w & 2w \\ 2w & -4w \end{matrix} \right), \quad \left( \begin{matrix} w \\ 2w \end{matrix} \right) \]

    with $w$ arbitrary and $m_1 \geq m_2$ and $2m_2 \geq m_1$, or

  3. are given by

    \[ \left( \begin{matrix} m_1 \\ m_2 \end{matrix} \right), \quad \left( \begin{matrix} -2w & 3w \\ 3w & -6w \end{matrix} \right), \quad \left( \begin{matrix} w \\ 3w \end{matrix} \right) \]

    with $w$ arbitrary and $2m_1 \geq 3m_2$ and $2m_2 \geq m_1$.

Proof. See discussion above. $\square$

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