Let $\mathcal{A}\_{g,D}$ be the moduli space of abelian varieties of dimension $g$ and polarization $D$ of type $(d\_1, \ldots, d\_g)$.


Let $\mathcal{M}$ be the moduli space parametrizing pairs $(A, \mathcal{L})$, where $A \in \mathcal{A}\_{g,D}$ and $\mathcal{L}$ is a non-trivial $2$-torsion line bundle on $A$, i.e. a non-zero element of $\textrm{Pic}^0(A)$ such that $\mathcal{L}^{\otimes 2}=\mathcal{O}\_A$.


Then there is a covering


$\pi \colon \mathcal{M} \longrightarrow \mathcal{A}\_{g,D}$


of degree $2^{2g}-1$, given by $[A, \mathcal{L}] \to A$.


**Question** Is the monodromy group of $\pi$ transitive? Or, equivalently, is $\mathcal{M}$ connected?


The answer is *yes* when $g=1$, i.e. for elliptic curves. In fact in this case $\mathcal{M}$ is a particular case of a more general construction called the *moduli space of spin curves*, which was studied by several authors (Cornalba, Verra, Farkas, etc). 


What about the case $g \geq 2$? Is there any reference? I'm particularly interested to the case where $g=2$ and $D=(1,2)$. 


**EDIT** Let me explain better the case I'm interested in, hoping that this can be helpful. Let $(A, D)$ be an abelian surface with polarization of type $(1,2)$, which I assume to be not of product type. The linear system $|D|$ is a pencil, that is $h^0(D)=2$, its general element is irreducible and up to a translation we can take $\mathcal{O}\_A(D)$ symmetric, i.e. $(-1)\_A^\* \mathcal{O}\_A(D)= \mathcal{O}\_A(D)$.
Therefore the base locus of $|D|$ is given by the zero element $o$ of $A$ and by three $2$-division points $e\_1$, $e\_2$, $e\_3$ such that $e\_1+e\_2=e\_3$.


There are exactly three $2$-torsion line bundles $\mathcal{L}\_1$, $\mathcal{L}\_2$, $\mathcal{L}\_3$ on $A$ such that there exists an element in the "translated pencil" $|D \otimes \mathcal{L}\_i|$ having a double point at $o$ (which is easily proven to be a node). The set 


$\{\mathcal{O}\_A, \mathcal{L}\_1, \mathcal{L}\_2, \mathcal{L}\_3\}$ 


form a subgroup of $\textrm{Pic}^0(A)$ isomorphic to $\mathbb{Z}/(2) \times \mathbb{Z}/(2)$, which is exactly the image of the map


$\phi \colon A[2] \longrightarrow \textrm{Pic}^0(A)[2]$,


see Laurent Moret-Bailly's answer.



$2$-torsion line bundles on abelian varieties
Let's stick to $D=(1,2)$. Associated to any point of $\mathcal{M}$ (say, a complex point) we have an abelian surface $A$ and an isogeny $\phi:A\to A'$ where $A'=\mathrm{Pic}^0(A)$ is the dual. The kernel of $\phi$, in our case, is isomorphic to $(\mathbb{Z}/2\mathbb{Z})^2$. So, the image of $A[2]$ in $A'$ is a canonically defined subgroup $H$ of $A'[2]$, isomorphic to $(\mathbb{Z}/2\mathbb{Z})^2$. On the other hand, the line bundle $\mathcal{L}$ is a nontrivial point in $A'[2]$. Clearly, those $(A,\phi, \mathcal{L})$ such that $\mathcal{L}\in H$ form a nontrivial open and closed subset of $\mathcal{M}$, so $\mathcal{M}$ is not connected.


