The Chow Ring Classes of ${\mathrm{PGL}}_3$ Orbit Closures in $\mathbb{G}(1,5)$

Abstract

The space of all pencils of conics in the plane $\mathbb{P}V$ (where $\dim V=3$) is a projective Grassmannian $\mathbb{G}(1,\mathbb{P}{\mathrm{Sym}}^2{V}^{\ast })$ and admits a natural $\mathrm{PGL}(V)$ action. It is a classical theorem that this action has exactly eight orbits, and in fact that the orbit of a pencil $\ell \subset \mathbb{P}{\mathrm{Sym}}^2{V}^{\ast }$ is determined completely by its position with respect to the Veronese surface $X\subset \mathbb{P}{\mathrm{Sym}}^2{V}^{\ast }$ of rank 1 conics and its secant variety $S(X)\subset \mathbb{P}{\mathrm{Sym}}^2{V}^{\ast }$, which is the cubic fourfold of rank 2 conics. In this paper, we present some geometric descriptions of these orbits. Then, using a mixture of direct enumerative techniques and some Chern class computations, we present a calculation of the classes of the orbit closures in the Chow ring of this Grassmannian (and consequently also of their degrees under the Plücker embedding $\mathbb{G}(1,\mathbb{P}{\mathrm{Sym}}^2{V}^{\ast })\hookrightarrow \mathbb{P}{\Lambda }^2{\mathrm{Sym}}^2{V}^{\ast }$).

26 pages, 2 figures, 1 table