Complete intersection quiver settings with one dimensional vertices

TL;DR

This paper characterizes a class of quiver settings with one-dimensional vertices whose semi-simple representations form a complete intersection variety, providing reduction techniques and identifying forbidden subquivers.

Contribution

It introduces a reduction method to analyze quiver settings with one-dimensional vertices and characterizes the class by forbidden subquivers, extending previous results.

Findings

Class of quivers with complete intersection semi-simple representations identified

Reduction to a single vertex quiver via combinatorial steps demonstrated

Characterization by absence of specific subquivers established

Abstract

We describe the class of quiver settings with one dimensional vertices whose semi-simple representations are parametrized by a complete intersection variety. We show that these quivers can be reduced to a one vertex quiver with some combinatorial reduction steps. We also show that this class consists of the quivers from which we can not obtain two specific non complete intersection quivers via contracting strongly connected components and deleting subquivers. The class of coregular quiver settings with arbitrary dimension vector, that has been described by an earlier result via reduction steps, can also be characterized by not containing a specific subquiver in the above sense.