What is the value of this hyperelliptic Hodge-type integral?

Question

Consider the moduli space $$ \overline{M}_{0,4}(B\mathbb{Z}/2) $$ This has virtual (and real) dimension one. In a certain sense this moduli space paramaterizes "genus 1 hyperelliptic curves"; that is, given a family of objects in this moduli space, by pulling back the universal family $pt \to B\mathbb{Z}/2$ we obtain a genus one curve with marked 2-torsion points. In particular, there is a morphism $$ \overline{M}_{0,4}(B\mathbb{Z}/2) \to \overline{M}_{1,4} $$ whose image is those genus 1 curves whose four marked points are exactly the 2-torsion points.

Because of this map, we can pull back the Hodge bundle to $\overline{M}_{0,4}(B\mathbb{Z}/2)$, and so we can consider the integral $$ \int_{\overline{M}_{0,4}(B\mathbb{Z}/2)} \lambda_1 $$

Question What is the value of this integral?

Some thoughts: We know that $\int_{\overline{M}_{1,1}} \lambda_1 = \frac{1}{24}$. Since the Hodge bundle pulls back via the forgetful maps (forgetting marked points), it seems that we should end up with either $3!\frac{1}{24} = \frac{1}{4}$ (choosing our favourite marked point, mapping to $\overline{M}_{1,1}$) or $4!\frac{1}{24} = 1$ (forgetting all the marked points).

I suppose another way of phrasing this: What is the degree of the map $\overline{M}_{0,4}(B\mathbb{Z}/2) \to \overline{M}_{1,1}$?

Answer 1

You are right that it's equivalent to compute the degree of the forgetful map. The degree is $3!$, since it's the quotient by $S_3$ permuting the last three markings. If you divide by $S_4$ you get the moduli stack of genus one curves with a distinguished degree $2$ map to $\mathbf P^1$, which is not the same as $\overline M_{1,1}$.

In fact there is a canonical isomorphism of stacks $\overline M_{0,4}(B\mathbf Z/2) \cong X(2)$, where $X(2)$ is the modular curve for the full level 2 congruence subgroup, parametrizing elliptic curves with a basis of their 2-torsion. This is easy to show on the interior. To get the result also on the boundary, use the modular interpretation of the cusps of $X(2)$ described in Deligne--Rapoport. For any $T \to X(2)$, you get a family $E \to T$ of generalized elliptic curves, and inversion in the group structure defines an admissible double cover $E \to E/\langle \pm 1\rangle$ branched over four points where $E/\langle \pm 1\rangle$ is a rational curve, hence a $T$-point of $\overline M_{0,4}(B\mathbf Z/2)$.