Families of Curves Wanted

An interesting problem

Let \(n\) be a large positive integer.  Recently I've been looking for a family of curves \(f_n: \mathscr{C}_n\to \mathbb{P}^1\) with the following properties:

  • \(f_n\) is flat and proper of relative dimension \(1\),
  • the general fiber of \(f_n\) is smooth, and the family is not isotrivial
  • every singularity that appears in a fiber of \(f_n\) is etale-locally of the form $$xy=t^n$$ where \(t\) is a parameter on \(\mathbb{P}^1\).

Please let me know if you know of such a family; another way of saying this is that I am looking for a rational curve in \(\overline{\mathscr{M}_{g}}\) such that every point of tangency with the boundary has order \(n\).  Nicola Tarasca has suggested a deformation-theoretic construction, which is promising; as I understand it, his idea is to construct a family of admissible covers with the desired singularities (where the general member of the family is not smooth, but instead has rational components, and then try to smooth it while preserving the desired singularities).  I have not checked to see if this works yet, but it seems like a reasonable idea.

Why am I interested in this?

Let \(U\subset \mathbb{P}^1\) be the Zariski-open subset over which \(f_n\) is smooth.  Then I claim that the monodromy representation $$\pi_1(U, x)\to GL_n(H^1(\mathscr{C}_{n,x}, \mathbb{Z}))$$ is trivial mod \(n\).  In general, it's quite hard to construct such representations, as I show for example in my paper Arithmetic Restrictions on Geometric Monodromy.  My interest in this question comes in part from wanting to know if the results in that paper are sharp.

More generally, the geometric torsion conjecture predicts that there exists an integer \(N=N(g)\) such that if \(A\) is a tracless \(g\)-dimensional Abelian variety over \(\mathbb{C}(t)\), then $$\#|A(\mathbb{C}(t))_{\text{tors}}|<N.$$  This conjecture is open -- moreover, to my knowledge, we don't know for sure if \(N\) has to depend on \(g\)!  The family of Jacobians of a family of curves as above would give an example showing that \(N\) must indeed depend on \(g\).  The interesting examples I know of Abelian varieties over \(\mathbb{C}(t)\) with lots of torsion come from genus zero Shimura curves, but there are only finitely many of those.

Let me briefly sketch the proof that a family of curves as above gives the desired example.  The idea is that the local monodromy associated to the singularity $$xy=t^n$$ has the form $$\begin{pmatrix} 1 & n\\ 0 & 1\end{pmatrix},$$ which is equal to the identity mod \(n\).  So for such a family of curves, the local monodromy at each point should vanish mod \(n\).  But the fundamental group of a punctured genus zero curve is generated by inertia, so we're done.