Is it true that for any $g\geq 1$ and $n$ such that $\overline{M}\_{g,n}$ has dimension at least two the locus in $\overline{M}\_{g,n}$ parametrizing reducible curves which are union of an elliptic tails with $6$ automorphisms and a curve of genus $g-1$ is a locus of non canonical singularities? Could it happen that for small values of $g$ these singularities are canonical?



Non canonical singularities of moduli spaces of curves
We denote by $\Delta\_{1}$ the image in $\overline{M}\_{g,n}$ of $\overline{M}\_{1,1}\times \overline{M}\_{g-1,n+1}$, the divisor parametrizing curves with elliptic tails. 


Let $Z\_6$ the image of $[E\_6]\times \overline{M}\_{g-1,n+1}\subset \Delta\_{1}$, the codimension two loci where the elliptic tail has six automorphisms.


For any $g\geq 1$ the sub-variety $Z\_6$ is a locus of codimension three non-canonical singularities. Indeed it is a locus of $\frac{1}{3}(1,1)$ transversal singularities. 


