Minimal exponents of hyperplane sections: a conjecture of Teissier

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 ${\tilde{\alpha }}_P(f)\ge {\tilde{\alpha }}_P(f{|}_H)+\frac{1}{{\theta }_P(f)+1}$, where ${\tilde{\alpha }}_P(f)$ and ${\tilde{\alpha }}_P(f{|}_H)$ are the minimal exponents at $P$ of $f$ and $f{|}_H$, 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 (1984) and Elduque–Mustaţă (2021). 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 ${\tilde{\alpha }}_P(f)\le {\tilde{\alpha }}_P(f{|}_H)+\frac{1}{{\mathrm{mult}}_P(f)}$, extending a result of Loeser from the case of isolated singularities.