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? 
 A: 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.    
A: 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
A: 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.
