Given an oriented manifold and its immersion in a euclidean space, we compute the oriented cobordism class of the manifold of [sum ]1r singular points of the projection of the immersion to a hyperplane. For immersions of non-oriented manifolds, we show that the cobordism class of the domain manifold determines those of all [sum ]1r singularity manifolds of the hyperplane projection. Finally, we investigate the possible (algebraic) number of cusps (that is, [sum ]1,1 singular points) of generic maps of oriented 4t-manifolds in R6t−1.