  We classify reflective automorphic products of singular weight under certain
regularity assumptions. Using obstruction theory we show that there are exactly
11 such functions. They are naturally related to certain conjugacy classes in
Conway's group $\text{Co}_0$. The corresponding modular varieties have a very
rich geometry. We establish a bijection between their $1$-dimensional type-$0$
cusps and the root systems in Schellekens' list. We also describe a
$1$-dimensional cusp along which the restriction of the automorphic product is
given by the eta product of the corresponding class in $\text{Co}_0$. Finally
we apply our results to give a complex-geometric proof of Schellekens' list.
