On the codimension 1 PGL(3) orbit closures in $\text{Gr}(3,6)$

TL;DR

This paper studies the orbit closures of the PGL(3) action on the Grassmannian of 3-dimensional subspaces of conics, specifically calculating the classes of codimension 1 orbit closures in the Chow ring.

Contribution

It explicitly computes the classes of codimension 1 orbit closures under PGL(3) action on Gr(3, V_2), extending prior understanding of orbit structure.

Findings

Identified the entire one-parameter family of codimension 1 orbits.

Determined the classes of these orbit closures in the Chow ring.

Included two special orbits among the codimension 1 cases.

Abstract

The projective linear group $\text{PGL}(3)$ naturally acts on the Grassmannian $\text{Gr}(3, V_2)$ of $3$-dimensional subspaces of the vector space $V_2$ of homogeneous conics in 3 variables. It was proved by Abdallah, Emsalem and Iarrobino in 2021 that this action has a one-parameter family of orbits along with 14 special orbits. The codimension 1 orbits of this action consist of the entire one-parameter family of orbits, along with 2 of the 14 special orbits. In this paper, we calculate the classes of the codimension 1 orbit closures in the Chow ring of $\text{Gr}(3, V_2)$.