Arc diagram varieties

TL;DR

This paper studies the geometric structure of orbit stratifications in algebraic group actions on short exact sequences of nilpotent operators, linking combinatorial arc diagrams with algebraic geometry.

Contribution

It introduces a new combinatorial approach using arc diagrams and an extended bubble sort algorithm to analyze orbit degenerations in nilpotent operator sequences.

Findings

Orbit stratification is governed by arc diagram combinatorics.

Chains of orbits with dimension difference one are constructed.

The approach connects geometric properties with combinatorial algorithms.

Abstract

Let $k$ be an algebraically closed field and $\alpha$, $\beta$, $\gamma$ be partitions. An algebraic group acts on the constructible set of short exact sequences of nilpotent $k$-linear operators of Jordan types $\alpha$, $\beta$, and $\gamma$, respectively; we are interested in the stratification given by the orbits in the case where all parts of $\alpha$ are at most 2. Geometric properties of the degeneration relation are controlled by the combinatorics of arc diagrams. The extended bubble sort algorithm is used to construct chains of orbits such that subsequent strata have dimension difference equal to one.