Title: 
The Genus of Moduli Spaces of $(n_1,n_2,n_3,n_4)$-Curves

Abstract:  
Following up on my most recent talk, I will review what $(n_1,n_2,n_3,n_4)$-curves are and present the conjecture I am trying to prove as my PhD-project, namely:

If $0<n_1<n_2<n_3<n_4$ are such that $(n_1,n_2,n_3,n_4)$ is admissible and $n_4$ is sufficiently large, 
then $\dim(\mathcal{M}(n_1,n_2,n_3,n_4))=1$. Furthermore, the genus of the moduli space curve $\mathcal{M}(n_1,n_2,n_3,n_4)$ tends to infinity with $n_4$.

In particular, I will be talking about getting "bad factors", where the discrimant of the universal curve equation vanishes, under control.