Spectrum of plane curves via knot theory

TL;DR

This paper employs topological methods, including knot theory invariants, to analyze the spectral properties of plane algebraic curves and polynomials, establishing semicontinuity results and relating spectra at infinity to singular points.

Contribution

It introduces a topological approach using Seifert forms and signatures to study spectra of plane curves and polynomials, providing new proofs and insights into their semicontinuity properties.

Findings

Reproved semicontinuity of spectrum at infinity using knot invariants.

Linked spectra at infinity with spectra of singular points of fibers.

Extended understanding of spectral behavior of plane algebraic curves.

Abstract

We use topological methods to study various semicontinuity properties of spectra of singular points of plane algebraic curves and of polynomials in two variables at infinity. Using Seifert forms and the Tristram--Levine signatures of links, we reprove (in a slightly weaker version) a result obtained by Steenbrink and Varchenko on semicontinuity of spectrum at infinity. We also relate the spectrum at infinity of a polynomial with spectra of singular points of a chosen fiber.