PGL orbits in tree varieties

TL;DR

This paper introduces tree varieties as a generalization of partial flag varieties, studies PGL orbits on them, and characterizes conditions for finitely many orbits and dense orbits, extending classical results.

Contribution

It generalizes the classification of PGL orbits to tree varieties and provides criteria for dense orbits, broadening understanding of orbit structures in algebraic geometry.

Findings

Characterization of tree varieties with finitely many PGL orbits.

Criteria for the existence of dense PGL orbits in tree varieties.

Identification of conditions for dense orbits in two-step flag varieties.

Abstract

In this paper, we introduce tree varieties as a natural generalization of products of partial flag varieties. We study orbits of the PGL action on tree varieties. We characterize tree varieties with finitely many PGL orbits, generalizing a celebrated theorem of Magyar, Weyman and Zelevinsky. We give criteria that guarantee that a tree variety has a dense PGL orbit and provide many examples of tree varieties that do not have dense PGL orbits. We show that a triple of two-step flag varieties $F(k_1, k_2; n)^3$ has a dense PGL orbit if and only if $k_1 + k_2 \not= n$.