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55.6 Classification of minimal type for genus zero and one

The title of the section explains it all.

Lemma 55.6.1 (Genus zero). The only minimal numerical type of genus zero is $n = 1$, $m_1 = 1$, $a_{11} = 0$, $w_1 = 1$, $g_1 = 0$.

Lemma 55.6.2 (Genus one). The minimal numerical types of genus one are up to equivalence

  1. $n = 1$, $a_{11} = 0$, $g_1 = 1$, $m_1, w_1 \geq 1$ arbitrary,

  2. $n = 2$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

    \[ \left( \begin{matrix} m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & 2w \\ 2w & -2w \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \end{matrix} \right) \]

    with $w$ and $m$ arbitrary,

  3. $n = 2$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

    \[ \left( \begin{matrix} 2m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & 4w \\ 4w & -8w \end{matrix} \right), \quad \left( \begin{matrix} w \\ 4w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \end{matrix} \right) \]

    with $w$ and $m$ arbitrary,

  4. $n = 3$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

    \[ \left( \begin{matrix} m \\ m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & w \\ w & -2w & w \\ w & w & -2w \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \end{matrix} \right) \]

    with $w$ and $m$ arbitrary,

  5. $n = 3$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

    \[ \left( \begin{matrix} m \\ 2m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & 0 \\ w & -2w & 3w \\ 0 & 3w & -6w \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ 3w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \end{matrix} \right) \]

    with $w$ and $m$ arbitrary,

  6. $n = 3$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

    \[ \left( \begin{matrix} m \\ 2m \\ 3m \end{matrix} \right), \quad \left( \begin{matrix} -6w & 3w & 0 \\ 3w & -6w & 3w \\ 0 & 3w & -2w \end{matrix} \right), \quad \left( \begin{matrix} 3w \\ 3w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \end{matrix} \right) \]

    with $w$ and $m$ arbitrary,

  7. $n = 3$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

    \[ \left( \begin{matrix} 2m \\ 2m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & 2w & 0 \\ 2w & -4w & 4w \\ 0 & 4w & -8w \end{matrix} \right), \quad \left( \begin{matrix} w \\ 2w \\ 4w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \end{matrix} \right) \]

    with $w$ and $m$ arbitrary,

  8. $n = 3$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

    \[ \left( \begin{matrix} m \\ m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & 2w & 0 \\ 2w & -4w & 2w \\ 0 & 2w & -2w \end{matrix} \right), \quad \left( \begin{matrix} w \\ 2w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \end{matrix} \right) \]

    with $w$ and $m$ arbitrary,

  9. $n = 3$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

    \[ \left( \begin{matrix} m \\ 2m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -4w & 2w & 0 \\ 2w & -2w & 2w \\ 0 & 2w & -4w \end{matrix} \right), \quad \left( \begin{matrix} 2w \\ w \\ 2w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \end{matrix} \right) \]

    with $w$ and $m$ arbitrary,

  10. $n = 4$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

    \[ \left( \begin{matrix} m \\ m \\ m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & 0 & w \\ w & -2w & w & 0 \\ 0 & w & -2w & w \\ w & 0 & w & -2w \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right) \]

    with $w$ and $m$ arbitrary,

  11. $n = 4$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

    \[ \left( \begin{matrix} 2m \\ 2m \\ 2m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & 2w & 0 & 0 \\ 2w & -4w & 2w & 0 \\ 0 & 2w & -4w & 4w \\ 0 & 0 & 4w & -8w \end{matrix} \right), \quad \left( \begin{matrix} w \\ 2w \\ 2w \\ 4w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right) \]

    with $w$ and $m$ arbitrary,

  12. $n = 4$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

    \[ \left( \begin{matrix} m \\ m \\ m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & 2w & 0 & 0 \\ 2w & -4w & 2w & 0 \\ 0 & 2w & -4w & 2w \\ 0 & 0 & 2w & -2w \end{matrix} \right), \quad \left( \begin{matrix} w \\ 2w \\ 2w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right) \]

    with $w$ and $m$ arbitrary,

  13. $n = 4$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

    \[ \left( \begin{matrix} m \\ 2m \\ 2m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -4w & 2w & 0 & 0 \\ 2w & -2w & w & 0 \\ 0 & w & -2w & 2w \\ 0 & 0 & 2w & -4w \end{matrix} \right), \quad \left( \begin{matrix} 2w \\ w \\ w \\ 2w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right) \]

    with $w$ and $m$ arbitrary,

  14. $n = 4$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

    \[ \left( \begin{matrix} 2m \\ m \\ m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & w & 2w \\ w & -2w & 0 & 0 \\ w & 0 & -2w & 0 \\ 2w & 0 & 0 & -4w \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ w \\ 2w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right) \]

    with $w$ and $m$ arbitrary,

  15. $n = 4$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

    \[ \left( \begin{matrix} 2m \\ m \\ m \\ 2m \end{matrix} \right), \quad \left( \begin{matrix} -4w & 2w & 2w & 2w \\ 2w & -4w & 0 & 0 \\ 2w & 0 & -4w & 0 \\ 2w & 0 & 0 & -2w \end{matrix} \right), \quad \left( \begin{matrix} 2w \\ 2w \\ 2w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right) \]

    with $w$ and $m$ arbitrary,

  16. $n = 5$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

    \[ \left( \begin{matrix} m \\ m \\ m \\ m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & 0 & 0 & w \\ w & -2w & w & 0 & 0 \\ 0 & w & -2w & w & 0 \\ 0 & 0 & w & -2w & w \\ w & 0 & 0 & w & -2w \\ \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ w \\ w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right) \]

    with $w$ and $m$ arbitrary,

  17. $n = 5$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

    \[ \left( \begin{matrix} m \\ 2m \\ 3m \\ 2m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & 0 & 0 & 0 \\ w & -2w & w & 0 & 0 \\ 0 & w & -2w & 2w & 0 \\ 0 & 0 & 2w & -4w & 2w \\ 0 & 0 & 0 & 2w & -4w \\ \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ w \\ 2w \\ 2w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right) \]

    with $w$ and $m$ arbitrary,

  18. $n = 5$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

    \[ \left( \begin{matrix} m \\ 2m \\ 3m \\ 4m \\ 2m \end{matrix} \right), \quad \left( \begin{matrix} -4w & 2w & 0 & 0 & 0 \\ 2w & -4w & 2w & 0 & 0 \\ 0 & 2w & -4w & 2w & 0 \\ 0 & 0 & 2w & -2w & w \\ 0 & 0 & 0 & w & -2w \\ \end{matrix} \right), \quad \left( \begin{matrix} 2w \\ 2w \\ 2w \\ w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right) \]

    with $w$ and $m$ arbitrary,

  19. $n = 5$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

    \[ \left( \begin{matrix} 2m \\ 2m \\ 2m \\ 2m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & 2w & 0 & 0 & 0 \\ 2w & -4w & 2w & 0 & 0 \\ 0 & 2w & -4w & 2w & 0 \\ 0 & 0 & 2w & -4w & 4w \\ 0 & 0 & 0 & 4w & -8w \\ \end{matrix} \right), \quad \left( \begin{matrix} w \\ 2w \\ 2w \\ 2w \\ 4w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right) \]

    with $w$ and $m$ arbitrary,

  20. $n = 5$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

    \[ \left( \begin{matrix} m \\ m \\ m \\ m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & 2w & 0 & 0 & 0 \\ 2w & -4w & 2w & 0 & 0 \\ 0 & 2w & -4w & 2w & 0 \\ 0 & 0 & 2w & -4w & 2w \\ 0 & 0 & 0 & 2w & -2w \\ \end{matrix} \right), \quad \left( \begin{matrix} w \\ 2w \\ 2w \\ 2w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right) \]

    with $w$ and $m$ arbitrary,

  21. $n = 5$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

    \[ \left( \begin{matrix} m \\ 2m \\ 2m \\ 2m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -4w & 2w & 0 & 0 & 0 \\ 2w & -2w & w & 0 & 0 \\ 0 & w & -2w & w & 0 \\ 0 & 0 & w & -2w & 2w \\ 0 & 0 & 0 & 2w & -4w \\ \end{matrix} \right), \quad \left( \begin{matrix} 2w \\ w \\ w \\ w \\ 2w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right) \]

    with $w$ and $m$ arbitrary,

  22. $n = 5$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

    \[ \left( \begin{matrix} 2m \\ m \\ m \\ m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & w & w & w \\ w & -2w & 0 & 0 & 0 \\ w & 0 & -2w & 0 & 0 \\ w & 0 & 0 & -2w & 0 \\ w & 0 & 0 & 0 & -2w \\ \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ w \\ w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right) \]

    with $w$ and $m$ arbitrary,

  23. $n = 5$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

    \[ \left( \begin{matrix} m \\ 2m \\ 2m \\ m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -4w & 2w & 0 & 0 & 0 \\ 2w & -2w & w & 0 & 0 \\ 0 & w & -2w & w & w \\ 0 & 0 & w & -2w & 0 \\ 0 & 0 & w & 0 & -2w \\ \end{matrix} \right), \quad \left( \begin{matrix} 2w \\ w \\ w \\ w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right) \]

    with $w$ and $m$ arbitrary,

  24. $n = 5$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

    \[ \left( \begin{matrix} 2m \\ 2m \\ 2m \\ m \\ m \end{matrix} \right), \quad \left( \begin{matrix} -2w & 2w & 0 & 0 & 0 \\ 2w & -4w & 2w & 0 & 0 \\ 0 & 2w & -4w & 2w & 2w \\ 0 & 0 & 2w & -4w & 0 \\ 0 & 0 & 2w & 0 & -4w \\ \end{matrix} \right), \quad \left( \begin{matrix} w \\ 2w \\ 2w \\ 2w \\ 2w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right) \]

    with $w$ and $m$ arbitrary,

  25. $n \geq 6$ and we have an $n$-cycle generalizing (16):

    1. $m_1 = \ldots = m_ n = m$,

    2. $a_{12} = \ldots = a_{(n - 1) n} = w$, $a_{1n} = w$, and for other $i < j$ we have $a_{ij} = 0$,

    3. $w_1 = \ldots = w_ n = w$

    with $w$ and $m$ arbitrary,

  26. $n \geq 6$ and we have a chain generalizing (19):

    1. $m_1 = \ldots = m_{n - 1} = 2m$, $m_ n = m$,

    2. $a_{12} = \ldots = a_{(n - 2) (n - 1)} = 2w$, $a_{(n - 1) n} = 4w$, and for other $i < j$ we have $a_{ij} = 0$,

    3. $w_1 = w$, $w_2 = \ldots = w_{n - 1} = 2w$, $w_ n = 4w$

    with $w$ and $m$ arbitrary,

  27. $n \geq 6$ and we have a chain generalizing (20):

    1. $m_1 = \ldots = m_ n = m$,

    2. $a_{12} = \ldots = a_{(n - 1) n} = w$, and for other $i < j$ we have $a_{ij} = 0$,

    3. $w_1 = w$, $w_2 = \ldots = w_{n - 1} = 2w$, $w_ n = w$

    with $w$ and $m$ arbitrary,

  28. $n \geq 6$ and we have a chain generalizing (21):

    1. $m_1 = w$, $w_2 = \ldots = m_{n - 1} = 2m$, $m_ n = m$,

    2. $a_{12} = 2w$, $a_{23} = \ldots = a_{(n - 2) (n - 1)} = w$, $a_{(n - 1) n} = 2w$, and for other $i < j$ we have $a_{ij} = 0$,

    3. $w_1 = 2w$, $w_2 = \ldots = w_{n - 1} = w$, $w_ n = 2w$

    with $w$ and $m$ arbitrary,

  29. $n \geq 6$ and we have a type generalizing (23):

    1. $m_1 = m$, $m_2 = \ldots = m_{n - 3} = 2m$, $m_{n - 1} = m_ n = m$,

    2. $a_{12} = 2w$, $a_{23} = \ldots = a_{(n - 2) (n - 1)} = w$, $a_{(n - 2) n} = w$, and for other $i < j$ we have $a_{ij} = 0$,

    3. $w_1 = 2w$, $w_2 = \ldots = w_ n = w$

    with $w$ and $m$ arbitrary,

  30. $n \geq 6$ and we have a type generalizing (24):

    1. $m_1 = \ldots = m_{n - 3} = 2m$, $m_{n - 1} = m_ n = m$,

    2. $a_{12} = \ldots = a_{(n - 2) (n - 1)} = 2w$, $a_{(n - 2) n} = 2w$, and for other $i < j$ we have $a_{ij} = 0$,

    3. $w_1 = w$, $w_2 = \ldots = w_ n = 2w$

    with $w$ and $m$ arbitrary,

  31. $n \geq 6$ and we have a type generalizing (22):

    1. $m_1 = m_2 = m$, $m_3 = \ldots = m_{n - 2} = 2m$, $m_{n - 1} = m_ n = m$,

    2. $a_{13} = w$, $a_{23} = \ldots = a_{(n - 2) (n - 1)} = w$, $a_{(n - 2) n} = w$, and for other $i < j$ we have $a_{ij} = 0$,

    3. $w_1 = \ldots = w_ n = w$,

    with $w$ and $m$ arbitrary,

  32. $n = 7$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

    \[ \left( \begin{matrix} m \\ 2m \\ 3m \\ m \\ 2m \\ m \\ 2m \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & 0 & 0 & 0 & 0 & 0 \\ w & -2w & w & 0 & 0 & 0 & 0 \\ 0 & w & -2w & 0 & w & 0 & w \\ 0 & 0 & 0 & -2w & w & 0 & 0 \\ 0 & 0 & w & w & -2w & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -2w & w \\ 0 & 0 & w & 0 & 0 & w & -2w \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ w \\ w \\ w \\ w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right) \]

    with $w$ and $m$ arbitrary,

  33. $n = 8$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

    \[ \left( \begin{matrix} m \\ 2m \\ 3m \\ 4m \\ 3m \\ 2m \\ m \\ 2m \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & 0 & 0 & 0 & 0 & 0 & 0 \\ w & -2w & w & 0 & 0 & 0 & 0 & 0 \\ 0 & w & -2w & w & 0 & 0 & 0 & 0 \\ 0 & 0 & w & -2w & w & 0 & 0 & w \\ 0 & 0 & 0 & w & -2w & w & 0 & 0 \\ 0 & 0 & 0 & 0 & w & -2w & w & 0 \\ 0 & 0 & 0 & 0 & 0 & w & -2w & 0 \\ 0 & 0 & 0 & w & 0 & 0 & 0 & -2w \\ \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ w \\ w \\ w \\ w \\ w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right) \]

    with $w$ and $m$ arbitrary,

  34. $n = 9$, and $m_ i, a_{ij}, w_ i, g_ i$ given by

    \[ \left( \begin{matrix} m \\ 2m \\ 3m \\ 4m \\ 5m \\ 6m \\ 4m \\ 2m \\ 3m \end{matrix} \right), \quad \left( \begin{matrix} -2w & w & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ w & -2w & w & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & w & -2w & w & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & w & -2w & w & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & w & -2w & w & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & w & -2w & w & 0 & w \\ 0 & 0 & 0 & 0 & 0 & w & -2w & w & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & w & -2w & 0 \\ 0 & 0 & 0 & 0 & 0 & w & 0 & 0 & -2w \\ \end{matrix} \right), \quad \left( \begin{matrix} w \\ w \\ w \\ w \\ w \\ w \\ w \\ w \\ w \end{matrix} \right), \quad \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix} \right) \]

    with $w$ and $m$ arbitrary.

Proof. This is proved in Section 55.5. See discussion at the start of Section 55.5. $\square$

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