Let $X$ be a compact Riemann surface (equivalently, a smooth projective curve over $\C$) of genus $g$, let $G$ be a group of automorphisms acting on $X$, and let $g_0$ be the genus of the quotient $Y:=X/G$. The natural projection $X \to Y$ is branched at $r$ points in $Y$, and the corresponding generators of the monodromy group have orders $m_1$, $m_2$, $\ldots$, $m_r$; the sequence of integers $[g_0; m_1, \ldots, m_r]$ is called the signature of the group action.

The label for the family of higher genus curves with a group $G \simeq$ SmallGroup$(n,d)$ acting on it with signature $[g_0; m_1, \ldots, m_r]$ is given as $$g.n\text{-}d.g_0.m_1\text{-}m_2\text{-} \cdots \text{-}m_r$$ For example, the genus 3 Hurwitz curve with automorphism group PSL$(2,7) \simeq $SmallGroup$(168,42)$ and signature $[0;2,3,7]$ is labeled: $$\text{5.168-42.0.2-3-7}$$ There may be several inequivalent actions described by that label, though. We also distinguish the actions by which conjugacy classes in $G$ the monodromy generators are from, creating passport labels. For our previous example $$\text{5.168-42.0.2-3-7.1} \text{ and } \text{5.168-42.0.2-3-7.2}$$ represent the two distinct actions of PSL$(2,7)$ as a Hurwitz group on a genus $3$ curve up to refined passports.

The suffixes $1$ and $2$ are ordinals that are assigned by lexicographically ordering the sequence of conjugacy class identifiers associated to a refined passport.

In order to explicitly identify elements of $G$ listed in generating vectors for a given refined passport (and elsewhere), we choose a particular permutation representation of $G$ as a subgroup of $S_{|G|}$, the symmetric group on $|G|$ elements (this choice depends only on the isomorphism class of $G$ and is the same for all $G$ with the same group identifier).

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- Review status: reviewed
- Last edited by Andrew Sutherland on 2018-07-03 16:01:18

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- 2018-07-03 16:01:18 by Andrew Sutherland (Reviewed)