Minimal exponents of hyperplane sections: a conjecture of Teissier

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 (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 \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.