The 'corrected Durfee's inequality' for homogeneous complete intersections

TL;DR

This paper investigates bounds on the singularity genus for isolated complete intersection singularities, shows the failure of Durfee's original inequality in higher codimension, and proposes a new, verified inequality for homogeneous cases.

Contribution

It demonstrates the failure of Durfee's inequality in higher codimension and introduces a new inequality that holds for homogeneous complete intersections.

Findings

Durfee's inequality fails for codimension > 1

A new inequality is proposed and verified for homogeneous cases

A combinatorial inequality underpins the new bound

Abstract

We address the conjecture of [Durfee1978], bounding the singularity genus, p_g, by a multiple of the Milnor number, \mu, for an n-dimensional isolated complete intersection singularity. We show that the original conjecture of Durfee, namely (n+1)!p_g\leq \mu, fails whenever the codimension r is greater than one. Moreover, we propose a new inequality, and we verify it for homogeneous complete intersections. In the homogeneous case the inequality is guided by a `combinatorial inequality', that might have an independent interest.