Some remarks about equations defining coincident root loci

TL;DR

This paper investigates the algebraic structure and singularities of varieties of binary forms with specified root multiplicity patterns, providing new generators, a criterion for smoothness, and filling a gap in existing proofs.

Contribution

It determines minimal local generators for the fiber product of these varieties and offers a new description of their singular locus and smoothness conditions.

Findings

Identified minimal local generators for the fiber product of $X____$ and its normalization.

Provided a new description of the singular locus of $X____$.

Established a criterion for the smoothness of $X____$.

Abstract

Consider the projective variety $X_\lambda$ of binary forms of degree $d$ whose linear factors are distributed according to the partition $\lambda$ of $d$. We determine minimal sets of local generators of the fiber product of $X_\lambda$ with its normalization, and we show that the local Jacobian matrices of this product 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 Chipalkatti's "On equations defining Coincident Root Loci". Also, we give a new description of the singular locus of $X_\lambda$ and a criterion for the smoothness of $X_\lambda$.