A universal discriminant formula for pencils of quadrics

Abstract

Let $V$ be a vector space of dimension $n+1$ over an algebraically closed field $\mathbb{k}$ of characteristic zero, and let $G_n = \operatorname{Gr}(2,\operatorname{Sym}^2V^\vee)$ be the Grassmannian parametrizing pencils of quadrics in $\mathbb{P}(V) \cong \mathbb{P}^n$. The determinant of the universal pencil defines a universal binary form of degree $n+1$. We prove that the divisor $\mathcal{D}_n\subseteq G_n$ of pencils whose determinant binary form has a multiple root has Chow class $[\mathcal{D}_n]=n(n+1)σ_1\in A^1(G_n),$ where $σ_1=c_1(S^\vee)$ and $S$ is the tautological rank-two subbundle on $G_n$. More generally, the higher-contact loci of determinant binary forms are computed by a universal jet formula. We also formulate the determinant-root collision strata as refined pullbacks of the universal collision strata for binary forms. For $n=3$, the main formula recovers the class $12σ_1$ for the boundary divisor in $\operatorname{Gr}(2,10)$ that the author established in a prior paper.