Durfee-type bound for some non-degenerate complete intersection singularities

TL;DR

This paper investigates bounds relating the Milnor number and singularity genus for certain non-degenerate complete intersection singularities, proving a corrected inequality in cases with large Newton diagrams.

Contribution

It establishes a new corrected bound for the Milnor number and singularity genus for Newton-non-degenerate complete intersection singularities with large Newton diagrams.

Findings

Proved the corrected inequality for complete intersections.

Established a stronger inequality for hypersurface cases.

Verified the bounds for cases with large Newton diagrams.

Abstract

The Milnor number, \mu(X,0), and the singularity genus, p_g(X,0), are fundamental invariants of isolated hypersurface singularities (more generally, of local complete intersections). The long standing Durfee conjecture (and its generalization) predicted the inequality \mu(X,0) \geq (n+1)!p_g(X,0), here n=dim(X,0). Recently we have constructed counterexamples, proposed a corrected bound and verified it for the homogeneous complete intersections. In the current paper we treat the case of germs with Newton-non-degenerate principal part when the Newton diagrams are "large enough", i.e. they are large multiples of some other diagrams. In the case of local complete intersections we prove the corrected inequality, while in the hypersurface case we prove an even stronger inequality.