Weight two compactly supported cohomology of moduli spaces of curves

We study the weight 2 graded piece of the compactly supported rational cohomology of the moduli spaces of curves $M_{g,n}$ and show that this can be computed as the cohomology of a graph complex that is closely related to graph complexes arising in the study of embedding spaces. For $n = 0$, we express this cohomology in terms of the weight zero compactly supported cohomology of $M_{g',n'}$ for $g' \leq g$ and $n' \leq 2$, and thereby produce several new infinite families of nonvanishing unstable cohomology groups on $M_g$, including the first such families in odd degrees. In particular, we show that the dimension of $H^{4g-k}(M_g)$ grows at least exponentially with $g$, for $k \in \{ 8, 9, 11, 12, 14, 15, 16, 18, 19 \}$.