# Monodromy group of 1-dimensional families of hyperelliptic curves

If $f: \mathcal{C} \rightarrow \mathcal{P}_{2g+2}$ is a general family of hyperelliptic curves (defined over $\mathbb{C}$), we know that the algebraic connected monodromy group $Mon^{0}$ of this family is equal to the full symplectic group. Now if we have a 1-parameter family of hyperelliptic curves given for example by the expression

$y^{2}= (x-h_{1}(t))....(x-h_{2g+2}(t))$ where $t$ is the parameter in $\mathbb{P}^{1}$ and the $h_{i}$ are holomorphic functions in $t$, is the group $Mon^{0}$ of this family also equal to the full symplectic group $Sp_{2g+2}$? if not, how can one compute the monodromy group of such family?

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Not in general. The monodromy group could even be trivial (e.g. if each $h_m(t)$ is a constant function, or of the form $c_m t^{2m}$ for some constants $c_m$). Or if each $h_{2m+1}(t) = -h_{2m+2}(t)$ the monodromy must commute with the involution $(x,y) \leftrightarrow (-x,y)$. I don't know if there's an algorithm that can be guaranteed to compute the monodromy in all cases. –  Noam D. Elkies Nov 10 '12 at 17:36
Thanks. I did not understand the reason why monodromy group is trivial if each $h_{m}$ is of the form $c_{m}t^{2m}$. Also by the involution you probably mean the hyperelliptic involution $(x,y)→(x,−y)$? –  Jack Nov 10 '12 at 17:51
When $h_m = c_m t^{2m}$, all nonzero choices of $t$ yield isomorphic curves. When each $h_{2m+1}(t) = -h_{2m+2}(t)$, the right-hand side of the defining equation $y^2 = \prod_i (x-h_i(t))$ is a polynomial in $x^2$ so the curve has an additional involution (the hyperelliptic involution $(x,y) \leftrightarrow (x,-y)$ must commute with the monodromy as well, but this doesn't help because that involution multiplies all the holomorphic differentials by $-1$). –  Noam D. Elkies Nov 10 '12 at 17:54

I believe that one must specify the base (in this case, the open subset of ${\mathbb P}^1$ over which the parameter $t$ varies) before talking about monodromy. If I understand correctly, the monodromy is the image of the action of the fundamental group of the base on the cohomology of the fibre.

If the base is all of ${\mathbb P}^1$ or the complex plane, then the monodromy is trivial (and hence so is the identity component of the Zariski closure).

If the base is ${\mathbb C}^*$, then the monodromy is a cyclic group, and hence $Mon ^0$ will be an abelian subgroup of the symplectic group.

If the image of the base in ${\mathcal P}_{2g+1}$ is such that at the level of fundamental groups, the map is surjective, then the monodromy is indeed the full symplectic group.

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For this sort of discussion, the base should be taken to consist of all the nondegenerate fibers. –  Will Sawin Nov 19 '12 at 7:30
Yes, but the hard part is figuring out whether the map of fundamental groups is surjective! –  JSE Dec 22 '12 at 21:10
@JSE, you are absolutely right. What I had in mind was that the one dimensional family (the base) was a generic intersection of hyperplanes with the space ${\mathcal P}_{2g+1}$ so that by the homotopy version of Lefschetz' theorem, the map at fundamental groups is surjective. –  Venkataramana Dec 23 '12 at 2:03
Dear Aakumadula, could you clarify your meaning by "surjectivity at the level of fundamental groups?" because in the monodromy representation, there is only one fundamental group involved. Also, you mean if the base is the complement of hyperplanes, then the monodromy is the full symplectic group? –  Jack Dec 23 '12 at 22:18
Dear Jack, consider the monodromy action of the fundamental group of ${\mathcal P}_{2g+1}$ ( a point on ${\mathbb C}^{2g+1}$ on the first cohomology of the fibre, the hyperelliptic curve associated to the point. There is a theorem of A'campo which says that the image of this monodromy is of finite index in the integral symplectic group. In particular, the Zariski closure of the monodromy associated to ${\mathbb P}_{2g+1}$ is $Sp_{2g}$. –  Venkataramana Dec 24 '12 at 3:04

When you say "algebraic monodromy group" do you mean the etale fundamental group? In that case, you're really asking about the image of monodromy in the symplectic group mod p. At this point there are lots of techniques for showing monodromy is big; for instance, see Chris Hall's paper "Big symplectic or orthogonal monodromy modulo l"

http://arxiv.org/abs/math/0608718

where he proves, among other things, that a family of the form y^2 = f(x)(x-t) has big monodromy in this sense. That paper, the paper it cites, and the papers that cite it should give you good ideas about how to approach the computation of mod p monodromy for any particular family of hyperelliptic curves you might encounter.

On the other hand, you might have in mind the topological question about the monodromy of a family of holomorphic curves in the discrete group Sp(2g,Z), which is somewhat more subtle. It can be the case that the monodromy surjects onto Sp(2g,Z/p) for every p but has infinite index in Sp(2g,Z); this is called the thin case. Actually, for y^2 = f(x)(x-t) one can prove that the monodromy is big even in this stronger sense; this was first proved in unpublished work of JK Yu, and I give another argument in Example 5 of my expository paper "Superstrong Approximation in Monodromy Groups":

http://arxiv.org/abs/1210.3757

which also has some general discussion of thin and non-thin monodromy groups.

I learned the argument in that paper from Ian Agol as a result of asking an MO question about it:

The image of the point-pushing group in the hyperelliptic representation of the braid group

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It looks like "connected algebraic monodromy group" in OP's question means "connected component of Zariski closure". –  Misha Dec 23 '12 at 5:36

The following reference may be helpful to you:

[1] Yukio Matsumoto, José María Montesinos-Amilibia, Pseudo-periodic homeomorphisms and degeneration of Riemann Surfaces, Bull. Amer. Math. Soc., 30(1994),70

[2] M. Ishizaka, Classification of the periodic monodromies of hyperelliptic families, Nagoya, Math. J., 174(2004),187-199.

[3] M. Ishizaka, Monodromies of hyperelliptic families of genus three curves, Tohoku Math. J. (2), 56, no.1 (2004),1-26.

[4] M. Ishizaka, Presentation of hyperelliptic periodic monodromies and splitting families, Revista mathematica complutense, 20(2007), no.2,483-495.

[5] T. Arakawa, T. Ashikaga : Local splitting families of hyperelliptic pencils, I, Tohoku Math. J., 53(2001),369-394.

[6] T. Arakawa, T. Ashikaga : Local splitting families of hyperelliptic pencils, II,Nagoya Math.J.,175(2004),103-124.

[7] T. Ashikaga, H. Endo, Various aspects of degenerate families of Riemann surfaces, Sugaku Expositions, 19(2006),171-196.

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