Triple Root Systems, Quasi-determinantal Quivers and Linear Free Divisors

TL;DR

This paper introduces a new root system for rational triple singularities, linking it to linear free divisors, and extends the framework to rational quasi-determinantal singularities, enriching the understanding of their algebraic and geometric properties.

Contribution

It constructs a new root system for rational triple singularities, establishes their connection to linear free divisors, and generalizes the results to rational quasi-determinantal singularities.

Findings

New root system for rational triple singularities

Linear free divisors associated with each root

Generalization to rational quasi-determinantal singularities

Abstract

We start by constructing a new root system for rational triple singularities and determine the number of roots for each rational triple singularity. Then we show that, for each root, we obtain a linear free divisor. So we obtain a new family of linear free divisors. This gives the converse part of an existing theorem which says, by using the quiver representation, that linear free divisors come from a tree. We prove that our construction is independent of the orientation on the rational triple trees. Furthermore, we deduce that linear free divisors defined by rational triple quivers satisfy the logarithmic comparison theorem. In last section, we generalize the results of these results to rational quasi-determinantal singularities.