Tjurina spectrum and graded symmetry of missing spectral numbers

TL;DR

This paper investigates the Tjurina spectrum and graded symmetry of missing spectral numbers in hypersurface singularities, revealing their self-duality and deriving bounds related to spectral numbers.

Contribution

It establishes the canonical graded symmetry of missing spectral numbers and links this to the self-duality of the Jacobian ring, improving bounds in specific cases.

Findings

The difference of spectra has a canonical graded symmetry.

The number of missing spectral numbers below a threshold is bounded by rac{(\u03bc}-)2.

Improved estimate of Brianb4con-Skoda exponent in the semisimple monodromy case.

Abstract

For a hypersurface isolated singularity defined by a convergent power series $f$, the Steenbrink spectrum can be defined as the Poincar\'e polynomial of the graded quotients of the $V$-filtration on the Jacobian ring of $f$. The Tjurina subspectrum is defined by replacing the Jacobian ring with its quotient by the image of the multiplication by $f$. We prove that their difference (consisting of missing spectral numbers) has a canonical graded symmetry. This follows from the self-duality of the Jacobian ring, which is compatible with the action of $f$ as well as the $V$-filtration. It implies for instance that the number of missing spectral numbers which are smaller than $(n{+}1)/2$ (with $n$ the number of variables) is bounded by $[(\mu{-}\tau)/2]$. We can moreover improve the estimate of Brian\c{c}on-Skoda exponent in the semisimple monodromy case.