2020/08/24 by Dirks, Bradley, Mustata, Mircea
#14B05 #14F10 #32S25 #Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2008.10345
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 \widetildeαP(f)≥ \widetildeαP(f\vertH)+(1)/(θP(f)+1), where \widetildeαP(f) and \widetildeαP(f\vertH) are the minimal exponents at P of f and f\vertH, respectively, and θP(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 \widetildeαP(f)≤ \widetildeαP(f\vertH)+\frac1\rm multP(f), extending a result of Loeser from the case of isolated singularities.