Minimal exponents of hyperplane sections: a conjecture of Teissier (Q6064886)

Summary: 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 \(\tilde{\alpha} p(f) \geq \tilde{\alpha} p(f|_H) + \frac{1}{\theta p (f) + 1}\), 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 \(\tilde{\alpha} p(f) \leq \tilde{\alpha} p(f|_H) + \frac{1}{\operatorname{mult} p(f)}\), extending a result of Loeser from the case of isolated singularities.