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Hurwitz's theorem on automorphisms tells us that the group of automorphisms of a nonsingular complex algebraic curve of genus at least 2 is bounded above by $84(g-1)$ where $g$ is the genus of the curve. We also know that for infinitely many $g$, this bound is sharp.

My question is whether these curves are 'isolated' or not. More precisely, given a fixed genus $g$, can I find a a one parameter family of curves which achieve the maximal bound for the automorphism group.

Of course there are some $g$ for which the $84(g-1)$ bound is not achieved, but I am happy with a family of curves that have the largest occurring automorphism group for that genus. Along these lines, I would also be interested in some "highly symmetric" (large automorphism group) one parameter family of curves of given genus.

Any insight/references/ways-to-think-about-this would be of great help!

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    $\begingroup$ You may start by reading en.wikipedia.org/wiki/Hurwitz_surface and then realizing that the (2,3,7)-group cannot be deformed in PSL(2,R). (Equivalently, the configuration of 3 points on the 2-sphere is rigid.) $\endgroup$
    – Misha
    Commented May 24, 2013 at 13:26
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    $\begingroup$ The answer below tells you that Hurwitz curves are isolated, but if you look at this paper vgonzale.mat.utfsm.cl/publicaciones/symmetricpams.pdf, for instance, Rodriguez and Gonzalez find a one-parameter family of abelian varieties that have action of a symmetric group (and so it is quite a large group acting on the variety). When the dimension is $\leq 4$, the family consists entirely of Jacobians, and so we get one-parameter families of curves with action of the symmetric group. Not sure if this helps at all! $\endgroup$
    – rfauffar
    Commented May 24, 2013 at 17:57

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The proof of Hurwitz's theorem easily shows that any such curve is a cover of $\mathbb P^1$ ramified at only three points, where the degree can be computed as a function of the genus. Since there are only finitely many covers of $\mathbb P^1$ ramified over three points with bounded degree, there are only finitely many Hurwitz curves of each genus. Thus they do not have nontrivial families.

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  • $\begingroup$ Ah of course. The case that gives you the bound is when $X/G$ has genus zero. Should've seen that. Thank you! $\endgroup$
    – Dhruv
    Commented May 24, 2013 at 15:47
  • $\begingroup$ Here's an addendum to Will's answer to your question that might interest you. For any integer $g>1$, there are only finitely many curves $X$ of genus $g$ such that $X/ $ Aut$(X)$ is the projective line and the quotient map $X\to X/ $ Aut$(X)$ is ramified over only three points. (Same argument as Will's.) Such curves are called Galois Belyi curves, or Wolfart curves. Hurwitz curves are examples of Wolfart curves, but so are Fermat curves, and certain congruence modular curves. Anyway, the point I wanted to make is that such curves also cannot be put into a nontrivial family. $\endgroup$
    – Ariyan Javanpeykar
    Commented May 24, 2013 at 17:10
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Regarding your last question, if you find a surface that admits a branched cover to the sphere with $n > 3$ ramification points, then you can move $n-3$ of the ramification points to create a family of complex dimension $n-3$ with the same automorphism group.

One of the many equivalent definitions of quasiplatonic surfaces is that they are the surfaces with automorphism group of strictly locally maximal order in moduli space. These curves cover the sphere with 3 ramification points too, but the ramification indices are allowed to be other numbers than 2, 3, and 7.

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