Some remarks on equations defining coincident root loci

Let Xλ be the projective variety of binary forms of degree d whose linear factors are distributed according to the partition λ of d. We determine minimal sets of local generators of Xλ×Yλ, where Yλ is the normalization of Xλ, and we show that the local Jacobian matrices of Xλ×Yλ contain the product of the identity matrix of maximal rank with a unit. We use this to fill a gap in a crucial proof in Chipalkattiʼs “On equations defining Coincident Root Loci”. Also, we give a new description of the singular locus of Xλ and a criterion for the smoothness of Xλ.