Toric ideals of simple surface singularities

TL;DR

This paper investigates toric ideals associated with ADE surface singularities, providing explicit Gr"obner bases and minimal generators that reveal algebraic and combinatorial properties of these singularities.

Contribution

It introduces explicit Gr"obner bases for toric ideals from ADE trees, highlighting their minimal generators and squarefree initial ideals, advancing understanding of surface singularities.

Findings

Explicit Gr"obner bases with binomials of degree ≤ 4

Minimal generating sets for the toric ideals

Squarefree initial ideals obtained

Abstract

In this paper, we study a class of toric ideals obtained by using some geometric data of ADE trees which are the minimal resolution graphs of rational surface singularities. We compute explicit Gr\"obner bases for these toric ideals that are also minimal generating sets consisting of large number of binomials of degree $\leq 4$. In particular, they give rise to squarefree initial ideals as well.