PrIncipal quiver Grassmannians: conjectures

TL;DR

This paper explores the geometric properties of specific quiver Grassmannians associated with Dynkin quivers, based on extensive computational experiments, and proposes several conjectures about their structure.

Contribution

It introduces new conjectures on the algebraic geometry of principal quiver Grassmannians derived from Dynkin quivers, supported by computational evidence.

Findings

Formulated conjectures on geometric properties of these Grassmannians

Provided computational evidence supporting the conjectures

Identified patterns suggesting new algebraic structures

Abstract

Let $P$ and $I$ be a projective and an injective representations of a Dynkin quiver. We consider quiver Grassmannians of subrepresentations of dimension $\dim P$ inside representations of dimension $\dim P + \dim I$. Based on extensive computer experiments, we formulate several conjectures about the algebro-geometric properties of these quiver Grassmannians.