In the first part of this talk I will give an update on the connection between perfect ideals of codimension 3 and Schubert varieties of exceptional groups (and more generally opposite Schubert varieties for Kac-Moody groups associated to T-shaped graphs Tpqr). I will also discuss a parallel theory which points to similar connection between Gorenstein ideals of codimension 4 with n generators and opposite Schubert varieties in homogeneous spaces related to a Kac-Moody group of type En.