On equations defining coincident root loci

Coincident root loci of binary forms

Coincident root loci are subvarieties of $S^d(C^2)$--the space of binary forms of degree $d$--labelled by partitions of $d$. Given a partition $λ$, let $X_λ$ be the set of forms with root multiplicity corresponding to $λ$. There is a natural action of $GL_2(C)$ on $S^d(C^2)$ and the coincident root loci are invariant under this action. We calculate their equivariant Poincaré duals generalizing formulas of Hilbert and Kirwan. In the second part we apply these results to present the cohomology ring of the corresponding moduli spaces (stable points$/G$, semistable points$/G$, link of the singular…

Some remarks about equations defining coincident root loci

Consider the projective variety $X_λ$ 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 the fiber product of $X_λ$ 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 $X_λ$ and a criterion for the smoothness of $X_λ$.

重合根层的等变上同调与Plücker公式

Plücker formulas using equivariant cohomology of coincident root strata

本文提出利用等变上同调计算重合根轨迹的万有上同调类，证明广义Plücker公式为次数的多项式，并给出其次数上界及达到上界时的首项，展示了等变方法在枚举几何中的应用。

Algebraic boundaries among typical ranks for real binary forms of arbitrary degree

We show that the algebraic boundaries of the regions of real binary forms with fixed typical rank are always unions of dual varieties to suitable coincident root loci.

Singular loci of reflection arrangements and the containment problem

This paper provides insights into the role of symmetry in studying polynomial functions vanishing to high order on an algebraic variety. The varieties we study are singular loci of hyperplane arrangements in projective space, with emphasis on arrangements arising from complex reflection groups. We provide minimal sets of equations for the radical ideals defining these singular loci and study containments between the ordinary and symbolic powers of these ideals.

On the algebraic boundaries among typical ranks for real binary forms

We describe the algebraic boundaries of the regions of real binary forms with fixed typical rank and of degree at most eight, showing that they are dual varieties of suitable coincident root loci.