Minimal exponents of hyperplane sections: a conjecture of Teissier

TL;DR

This paper proves a conjecture by Teissier relating minimal exponents of a function with an isolated singularity and its restriction to a smooth hypersurface, introducing new techniques involving Hodge ideals and invariants.

Contribution

It establishes a new inequality connecting minimal exponents and a novel invariant, expanding understanding of singularity invariants and their behavior under hypersurface restrictions.

Findings

Proved Teissier's conjecture on minimal exponents.

Introduced new results on Hodge ideals and their behavior.

Extended inequalities to general hypersurfaces through multiplicity.

Abstract

We prove a conjecture of Teissier asserting that if $f$ has an isolated singularity at $P$ and $H$ is a smooth hypersurface through $P$, then $\widetilde{\alpha}_P(f)\geq \widetilde{\alpha}_P(f\vert_H)+\frac{1}{\theta_P(f)+1}$, where $\widetilde{\alpha}_P(f)$ and $\widetilde{\alpha}_P(f\vert_H)$ are the minimal exponents at $P$ of $f$ and $f\vert_H$, respectively, and $\theta_P(f)$ is an invariant obtained by comparing the integral closures of the powers of the Jacobian ideal of $f$ and of the ideal defining $P$. The proof builds on the approaches of Loeser and Elduque-Mustata. The new ingredients are a result concerning the behavior of Hodge ideals with respect to finite maps and a result about the behavior of certain Hodge ideals for families of isolated singularities with constant Milnor number. In the opposite direction, we show that for every $f$, if $H$ is a general hypersurface through $P$, then $\widetilde{\alpha}_P(f)\leq \widetilde{\alpha}_P(f\vert_H)+\frac{1}{{\rm mult}_P(f)}$, extending a result of Loeser from the case of isolated singularities.