On the Codimension-1 $\mathrm{PGL}_4$ Orbit Closures in $\mathrm{Gr}(2,10)$

Abstract

We study the natural action of $\mathrm{PGL}(V)$ on the Grassmannian $G=\operatorname{Gr}(2,\operatorname{Sym}^2 V^\vee)$, where $\dim V=4$ and points of $G$ are pencils of quadrics in $\mathbb{P}(V)\cong \mathbb{P}^3$. Here $\dim G=16$ while $\dim \mathrm{PGL}(V)=15$, so the generic orbit has codimension one and one expects a one-parameter family of generic orbits. We construct this family via the $j$-invariant of the discriminant binary quartic of a pencil. We then determine the codimension-one orbit closures and compute their Chow classes. The smooth codimension-one orbit closures are the reduced fibers of the $j$-map on the smooth locus, while the unique boundary divisor is the closure of the orbit of a nodal quartic complete intersection of arithmetic genus $1$ and geometric genus $0$. Every divisorial fiber of the rational $j$-map has class $12\sigma_1$ in $A^1(G)$. For the reduced codimension-one orbit closures one has $[\overline{O_a}]=12\sigma_1$ for $a\neq 0,1728,\infty$, $[\overline{O_{1728}}]=6\sigma_1$, $[\overline{O_0}]=4\sigma_1$, and $[T]=12\sigma_1$.