The Stacks project

Lemma 55.3.5. Let $n, m_ i, a_{ij}, w_ i, g_ i$ be a numerical type of genus $g$. If $n = 1$, then $a_{11} = 0$ and $g = 1 + m_1w_1(g_1 - 1)$. Moreover, we can classify all such numerical types as follows

  1. If $g < 0$, then $g_1 = 0$ and there are finitely many possible numerical types of genus $g$ with $n = 1$ corresponding to factorizations $m_1w_1 = 1 - g$.

  2. If $g = 0$, then $m_1 = 1$, $w_1 = 1$, $g_1 = 0$ as in Lemma 55.6.1.

  3. If $g = 1$, then we conclude $g_1 = 1$ but $m_1, w_1$ can be arbitrary positive integers; this is case (1) of Lemma 55.6.2.

  4. If $g > 1$, then $g_1 > 1$ and there are finitely many possible numerical types of genus $g$ with $n = 1$ corresponding to factorizations $m_1w_1(g_1 - 1) = g - 1$.

Proof. The lemma proves itself. $\square$

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